Application of Definite Integration Class 12 Maths 2 Miscellaneous Exercise 5 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 Questions and Answers.

12th Maths Part 2 Application of Definite Integration Miscellaneous Exercise 5 Questions And Answers Maharashtra Board

I. Choose the correct option from the given alternatives:

Question 1.
The area bounded by the region 1 ≤ x ≤ 5 and 2 ≤ y ≤ 5 is given by
(a) 12 sq units
(b) 8 sq units
(c) 25 sq units
(d) 32 sq units
Answer:
(a) 12 sq units

Question 2.
The area of the region enclosed by the curve y = \(\frac{1}{x}\), and the lines x = e, x = e2 is given by
(a) 1 sq unit
(b) \(\frac{1}{2}\) sq units
(c) \(\frac{3}{2}\) sq units
(d) \(\frac{5}{2}\) sq units
Answer:
(a) 1 sq unit

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 3.
The area bounded by the curve y = x3, the X-axis and the lines x = -2 and x = 1 is
(a) -9 sq units
(b) \(-\frac{15}{4}\) sq units
(c) \(\frac{15}{4}\) sq units
(d) \(\frac{17}{4}\) sq units
Answer:
(c) \(\frac{15}{4}\) sq units

Question 4.
The area enclosed between the parabola y2 = 4x and line y = 2x is
(a) \(\frac{2}{3}\) sq units
(b) \(\frac{1}{3}\) sq units
(c) \(\frac{1}{4}\) sq units
(d) \(\frac{3}{4}\) sq units
Answer:
(b) \(\frac{1}{3}\) sq units

Question 5.
The area of the region bounded between the line x = 4 and the parabola y2 = 16x is
(a) \(\frac{128}{3}\) sq units
(b) \(\frac{108}{3}\) sq units
(c) \(\frac{118}{3}\) sq units
(d) \(\frac{218}{3}\) sq units
Answer:
(a) \(\frac{128}{3}\) sq units

Question 6.
The area of the region bounded by y = cos x, Y-axis and the lines x = 0, x = 2π is
(a) 1 sq unit
(b) 2 sq units
(c) 3 sq units
(d) 4 sq units
Answer:
(d) 4 sq units

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 7.
The area bounded by the parabola y2 = 8x, the X-axis and the latus rectum is
(a) \(\frac{31}{3}\) sq units
(b) \(\frac{32}{3}\) sq units
(c) \(\frac{32 \sqrt{2}}{3}\) sq units
(d) \(\frac{16}{3}\) sq units
Answer:
(b) \(\frac{32}{3}\) sq units

Question 8.
The area under the curve y = 2√x, enclosed between the lines x = 0 and x = 1 is
(a) 4 sq units
(b) \(\frac{3}{4}\) sq units
(c) \(\frac{2}{3}\) sq units
(d) \(\frac{4}{3}\) sq units
Answer:
(d) \(\frac{4}{3}\) sq units

Question 9.
The area of the circle x2 + y2 = 25 in first quadrant is
(a) \(\frac{25 \pi}{3}\) sq units
(b) 5π sq units
(c) 5 sq units
(d) 3 sq units
Answer:
(a) \(\frac{25 \pi}{3}\) sq units

Question 10.
The area of the region bounded by the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is
(a) ab sq units
(b) πab sq units
(c) \(\frac{\pi}{a b}\) sq units ab
(d) πa2 sq units
Answer:
(b) πab sq units

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 11.
The area bounded by the parabola y2 = x and the line 2y = x is
(a) \(\frac{4}{3}\) sq units
(b) 1 sq unit
(c) \(\frac{2}{3}\) sq unit
(d) \(\frac{1}{3}\) sq unit
Answer:
(a) \(\frac{4}{3}\) sq units

Question 12.
The area enclosed between the curve y = cos 3x, 0 ≤ x ≤ \(\frac{\pi}{6}\) and the X-axis is
(a) \(\frac{1}{2}\) sq unit
(b) 1 sq unit
(c) \(\frac{2}{3}\) sq unit
(d) \(\frac{1}{3}\) sq unit
Answer:
(d) \(\frac{1}{3}\) sq unit

Question 13.
The area bounded by y = √x and line x = 2y + 3, X-axis in first quadrant is
(a) 2√3 sq units
(b) 9 sq units
(c) \(\frac{34}{3}\) sq units
(d) 18 sq units
Answer:
(b) 9 sq units

Question 14.
The area bounded by the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and the line \(\frac{x}{a}+\frac{y}{b}=1\) is
(a) (πab – 2ab) sq units
(b) \(\frac{\pi a b}{4}-\frac{a b}{2}\) sq units
(c) (πab – ab) sq units
(d) πab sq units
Answer:
(b) \(\frac{\pi a b}{4}-\frac{a b}{2}\) sq units

Question 15.
The area bounded by the parabola y = x2 and the line y = x is
(a) \(\frac{1}{2}\) sq unit
(b) \(\frac{1}{3}\) sq unit
(c) \(\frac{1}{6}\) sq unit
(d) \(\frac{1}{12}\) sq unit
Answer:
(c) \(\frac{1}{6}\) sq unit

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 16.
The area enclosed between the two parabolas y2 = 4x and y = x is
(a) \(\frac{8}{3}\) sq units
(b) \(\frac{32}{3}\) sq units
(c) \(\frac{16}{3}\) sq units
(d) \(\frac{4}{3}\) sq units
Answer:
(c) \(\frac{16}{3}\) sq units

Question 17.
The area bounded by the curve y = tan x, X-axis and the line x = \(\frac{\pi}{4}\) is
(a) \(\frac{1}{3}\) log 2 sq units
(b) log 2 sq units
(c) 2 log 2 sq units
(d) 3 log 2 sq units
Answer:
(a) \(\frac{1}{3}\) log 2 sq units

Question 18.
The area of the region bounded by x2 = 16y, y = 1, y = 4 and x = 0 in the first quadrant, is
(a) \(\frac{7}{3}\) sq units
(b) \(\frac{8}{3}\) sq units
(c) \(\frac{64}{3}\) sq units
(d) \(\frac{56}{3}\) sq units
Answer:
(d) \(\frac{56}{3}\) sq units

Question 19.
The area of the region included between the parabolas y2 = 4ax and x2 = 4ay, (a > 0) is given by
(a) \(\frac{16 a^{2}}{3}\) sq units
(b) \(\frac{8 a^{2}}{3}\) sq units
(c) \(\frac{4 a^{2}}{3}\) sq units
(d) \(\frac{32 a^{2}}{3}\) sq units
Answer:
(a) \(\frac{16 a^{2}}{3}\) sq units

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 20.
The area of the region included between the line x + y = 1 and the circle x2 + y2 = 1 is
(a) \(\frac{\pi}{2}-1\) sq units
(b) π – 2 sq units
(c) \(\frac{\pi}{4}-\frac{1}{2}\) sq units
(d) π – \(\frac{1}{2}\) sq units
Answer:
(c) \(\frac{\pi}{4}-\frac{1}{2}\) sq units

(II) Solve the following:

Question 1.
Find the area of the region bounded by the following curve, the X-axis and the given lines:
(i) 0 ≤ x ≤ 5, 0 ≤ y ≤ 2
(ii) y = sin x, x = 0, x = π
(iii) y = sin x, x = 0, x = \(\frac{\pi}{3}\)
Solution:
(i) Required area = \(\int_{0}^{5} y d x\), where y = 2
= \(\int_{0}^{5} 2 d x\)
= \([2 x]_{0}^{5}\)
= 2 × 5 – 0
= 10 sq units.

(ii) The curve y = sin x intersects the X-axis at x = 0 and x = π between x = 0 and x = π.
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q1(ii)
Two bounded regions A1 and A2 are obtained. Both the regions have equal areas.
∴ required area = A1 + A2 = 2A1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q1(ii).1

(iii) Required area = \(\int_{0}^{\pi / 3} y d x\), where y = sin x
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q1(iii)

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 2.
Find the area of the circle x2 + y2 = 9, using integration.
Solution:
By the symmetry of the circle, its area is equal to 4 times the area of the region OABO.
Clearly, for this region, the limits of integration are 0 and 3.
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q2
From the equation of the circle, y2 = 9 – x2.
In the first quadrant, y > 0
∴ y = \(\sqrt{9-x^{2}}\)
∴ area of the circle = 4 (area of the region OABO)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q2.1

Question 3.
Find the area of the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) using integration.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q3
By the symmetry of the ellipse, its area is equal to 4 times the area of the region OABO.
Clearly, for this region, the limits of integration are 0 and 5.
From the equation of the ellipse
\(\frac{y^{2}}{16}=1-\frac{x^{2}}{25}=\frac{25-x^{2}}{25}\)
∴ y2 = \(\frac{16}{25}\) (25 – x2)
In the first quadrant y > 0
∴ y = \(\frac{4}{5} \sqrt{25-x^{2}}\)
∴ area of the ellipse = 4(area of the region OABO)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q3.1

Question 4.
Find the area of the region lying between the parabolas:
(i) y2 = 4x and x2 = 4y
(ii) 4y2 = 9x and 3x2 = 16y
(iii) y2 = x and x2 = y.
Solution:
(i)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(i)
For finding the points of intersection of the two parabolas, we equate the values of y2 from their equations.
From the equation x2 = 4y, y = \(\frac{x^{2}}{4}\)
y = \(\frac{x^{4}}{16}\)
\(\frac{x^{4}}{16}\) = 4x
∴ x4 – 64x = 0
∴ x(x3 – 64) = 0
∴ x = 0 or x3 = 64 i.e. x = 0 or x = 4
When x = 0, y = 0
When x = 4, y = \(\frac{4^{2}}{4}\) = 4
∴ the points of intersection are 0(0, 0) and A(4, 4).
Required area = area of the region OBACO = [area of the region ODACO] – [area of the region ODABO]
Now, area of the region ODACO = area under the parabola y2 = 4x, i.e. y = 2√x between x = 0 and x = 4
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(i).1

(ii)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(ii)
For finding the points of intersection of the two parabolas, we equate the values of 4y2 from their equations.
From the equation 3x2 = 16y, y = \(\frac{3 x^{2}}{16}\)
∴ y = \(\frac{3 x^{4}}{256}\)
∴ \(\frac{3 x^{4}}{256}\) = 9x
∴ 3x4 – 2304x = 0
∴ x(x3 – 2304) = 0
∴ x = 0 or x3 = 2304 i.e. x = 0 or x = 4
When x = 0, y = 0
When x = 4, y = \(\frac{4^{2}}{4}\)
∴ the points of intersection are O(0, 0) and A(4, 4).
Required area = area of the region OBACO = [area of the region ODACO] – [area of the region ODABO]
Now, area of the region ODACO = area under the parabola y2 = 4x,
i.e. y = 2√x between x = 0 and x = 4
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(ii).1
Area of the region ODABO = area under the rabola x2 = 4y,
i.e. y = \(\frac{x^{2}}{4}\) between x = 0 and x = 4
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(ii).2

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

(iii)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(iii)
For finding the points of intersection of the two parabolas, we equate the values of y2 from their equations.
From the equation x2 = y, y = \(\frac{x^{2}}{y}\)
∴ y = \(\frac{x^{2}}{y}\)
∴ \(\frac{x^{2}}{y}\) = x
∴ x2 – y = 0
∴ x(x3 – y) = 0
∴ x = 0 or x3 = y
i.e. x = 0 or x = 4
When x = 0, y = 0
When x = 4, y = \(\frac{4^{2}}{4}\) = 4
∴ the points of intersection are O(0, 0) and A(4, 4).
Required area = area of the region OBACO = [area of the region ODACO] – [area of the region ODABO]
Now, area of the region ODACO = area under the parabola y2 = 4x,
i.e. y = 2√x between x = 0 and x = 4
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(iii).1
Area ofthe region ODABO = area under the rabola x2 = 4y,
i.e. y = \(\frac{x^{2}}{3}\) between x = 0 and x = 4
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q4(iii).2

Question 5.
Find the area of the region in the first quadrant bounded by the circle x2 + y2 = 4 and the X-axis and the line x = y√3.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q5
For finding the points of intersection of the circle and the line, we solve
x2 + y2 = 4 ………(1)
and x = y√3 ……..(2)
From (2), x2 = 3y2
From (1), x2 = 4 – y2
3y2 = 4 – y2
4y2 = 4
y2 = 1
y = 1 in the first quadrant.
When y = 1, r = 1 × √3 = √3
∴ the circle and the line intersect at A(√3, 1) in the first quadrant
Required area = area of the region OCAEDO = area of the region OCADO + area of the region DAED
Now, area of the region OCADO = area under the line x = y√3, i.e. y = \(\frac{x}{\sqrt{3}}\) between x = 0
and x = √3
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q5.1

Question 6.
Find the area of the region bounded by the parabola y2 = x and the line y = x in the first quadrant.
Solution:
To obtain the points of intersection of the line and the parabola, we equate the values of x from both equations.
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q6
∴ y2 = y
∴ y2 – y = 0
∴ y(y – 1) = 0
∴ y = 0 or y = 1
When y = 0, x = 0
When y = 1, x = 1
∴ the points of intersection are O(0, 0) and A(1, 1).
Required area = area of the region OCABO = area of the region OCADO – area of the region OBADO
Now, area of the region OCADO = area under the parabola y2 = x i.e. y = +√x (in the first quadrant) between x = 0 and x = 1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q6.1
Area of the region OBADO = area under the line y = x between x = 0 and x = 1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q6.2

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 7.
Find the area enclosed between the circle x2 + y2 = 1 and the line x + y = 1, lying in the first quadrant.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q7
Required area = area of the region ACBPA = (area of the region OACBO) – (area of the region OADBO)
Now, area of the region OACBO = area under the circle x2 + y2 = 1 between x = 0 and x = 1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q7.1
Area of the region OADBO = area under the line x + y = 1 between x = 0 and x = 1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q7.2
∴ required area = \(\left(\frac{\pi}{4}-\frac{1}{2}\right)\) sq units.

Question 8.
Find the area of the region bounded by the curve (y – 1)2 = 4(x + 1) and the line y = (x – 1).
Solution:
The equation of the curve is (y – 1)2 = 4(x + 1)
This is a parabola with vertex at A (-1, 1).
To find the points of intersection of the line y = x – 1 and the parabola.
Put y = x – 1 in the equation of the parabola, we get
(x – 1 – 1)2 = 4(x + 1)
∴ x2 – 4x + 4 = 4x + 4
∴ x2 – 8x = 0
∴ x(x – 8) = 0
∴ x = 0, x = 8
When x = 0, y = 0 – 1 = -1
When x = 8, y = 8 – 1 = 7
∴ the points of intersection are B (0, -1) and C (8, 7).
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q8
To find the points where the parabola (y – 1)2 = 4(x + 1) cuts the Y-axis.
Put x = 0 in the equation of the parabola, we get
(y – 1)2 = 4(0 + 1) = 4
∴ y – 1 = ±2
∴ y – 1 = 2 or y – 1 = -2
∴ y = 3 or y = -1
∴ the parabola cuts the Y-axis at the points B(0, -1) and F(0, 3).
To find the point where the line y = x – 1 cuts the X-axis.
Put y = 0 in the equation of the line, we get
x – 1 = 0
∴ x = 1
∴ the line cuts the X-axis at the point G (1, 0).
Required area = area of the region BFAB + area of the region OGDCEFO + area of the region OBGO
Now, area of the region BFAB = area under the parabola (y – 1)2 = 4(x + 1), Y-axis from y = -1 to y = 3
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q8.1
Since, the area cannot be negative,
Area of the region BFAB = \(\left|-\frac{8}{3}\right|=\frac{8}{3}\) sq units.
Area of the region OGDCEFO = area of the region OPCEFO – area of the region GPCDG
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q8.2
Since, area cannot be negative,
area of the region = \(\left|-\frac{1}{2}\right|=\frac{1}{2}\) sq units.
∴ required area = \(\frac{8}{3}+\frac{109}{6}+\frac{1}{2}\)
= \(\frac{16+109+3}{6}\)
= \(\frac{128}{6}\)
= \(\frac{64}{3}\) sq units.

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 9.
Find the area of the region bounded by the straight line 2y = 5x + 7, X-axis and x = 2, x = 5.
Solution:
The equation of the line is
2y = 5x + 7, i.e., y = \(\frac{5}{2} x+\frac{7}{2}\)
Required area = area of the region ABCDA = area under the line y = \(\frac{5}{2} x+\frac{7}{2}\) between x = 2 and x = 5
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q9

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5

Question 10.
Find the area of the region bounded by the curve y = 4x2, Y-axis and the lines y = 1, y = 4.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q10
By symmetry of the parabola, the required area is 2 times the area of the region ABCD.
From the equation of the parabola, x2 = \(\frac{y}{4}\)
In the first quadrant, x > 0
∴ x = \(\frac{1}{2} \sqrt{y}\)
∴ required area = \(\int_{1}^{4} x d y\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Miscellaneous Exercise 5 II Q10.1

Class 12 Maharashtra State Board Maths Solution 

Application of Definite Integration Class 12 Maths 2 Exercise 5.1 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 5 Application of Definite Integration Ex 5.1 Questions and Answers.

12th Maths Part 2 Application of Definite Integration Exercise 5.1 Questions And Answers Maharashtra Board

1. Find the area of the region bounded by the following curves, X-axis, and the given lines:

(i) y = 2x, x = 0, x = 5.
Solution:
Required area = \(\int_{0}^{5} y d x\), where y = 2x
= \(\int_{0}^{5} 2x d x\)
= \(\left[\frac{2 x^{2}}{2}\right]_{0}^{5}\)
= 25 – 0
= 25 sq units.

(ii) x = 2y, y = 0, y = 4.
Solution:
Required area = \(\int_{0}^{4} x d y\), where x = 2y
= \(\int_{0}^{4} 2 y d y\)
= \(\left[\frac{2 y^{2}}{2}\right]_{0}^{4}\)
= 16 – 0
= 16 sq units.

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1

(iii) x = 0, x = 5, y = 0, y = 4.
Solution:
Required area = \(\int_{0}^{5} y d x\), where y = 4
= \(\int_{0}^{5} 4 d x\)
= \([4 x]_{0}^{5}\)
= 20 – 0
= 20 sq units.

(iv) y = sin x, x = 0, x = \(\frac{\pi}{2}\)
Solution:
Required area = \(\int_{0}^{\pi / 2} y d x\), where y = sin x
= \(\int_{0}^{\pi / 2} \sin x d x\)
= \([-\cos x]_{0}^{\pi / 2}\)
= -cos \(\frac{\pi}{2}\) + cos 0
= 0 + 1
= 1 sq unit.

(v) xy = 2, x = 1, x = 4.
Solution:
For xy = 2, y = \(\frac{2}{x}\)
Required area = \(\int_{1}^{4} y d x\), where y = \(\frac{2}{x}\)
= \(\int_{1}^{4} \frac{2}{x} d x\)
= \([2 \log |x|]_{1}^{4}\)
= 2 log 4 – 2 log 1
= 2 log 4 – 0
= 2 log 4 sq units.

(vi) y2 = x, x = 0, x = 4.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q1 (vi)
The required area consists of two bounded regions A1 and A2 which are equal in areas.
For y2 = x, y = √x
Required area = A1 + A2 = 2A1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q1 (vi).1

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1

(vii) y2 = 16x, x = 0, x = 4.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q1 (vii)
The required area consists of two bounded regions A1 and A2 which are equal in areas.
For y2 = x, y = √x
Required area = A1 + A2 = 2A1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q1 (vii).1

2. Find the area of the region bounded by the parabola:

(i) y2 = 16x and its latus rectum.
Solution:
Comparing y2 = 16x with y2 = 4ax, we get
4a = 16
∴ a = 4
∴ focus is S(a, 0) = (4, 0)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q2 (i)
For y2 = 16x, y = 4√x
Required area = area of the region OBSAO
= 2 [area of the region OSAO]
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q2 (i).1

(ii) y = 4 – x2 and the X-axis.
Solution:
The equation of the parabola is y = 4 – x2
∴ x2 = 4 – y
i.e. (x – 0)2 = -(y – 4)
It has vertex at P(0, 4)
For points of intersection of the parabola with X-axis,
we put y = 0 in its equation.
∴ 0 = 4 – x2
∴ x2 = 4
∴ x = ± 2
∴ the parabola intersect the X-axis at A(-2, 0) and B(2, 0)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q2 (ii)
Required area = area of the region APBOA
= 2[area of the region OPBO]
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q2 (ii).1

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1

3. Find the area of the region included between:

(i) y2 = 2x and y = 2x.
Solution:
The vertex of the parabola y2 = 2x is at the origin O = (0, 0).
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (i)
To find the points of intersection of the line and the parabola, equaling the values of 2x from both the equations we get,
y2 = y
∴ y2 – y = 0
∴ y = 0 or y = 1
When y = 0, x = \(\frac{0}{2}\) = 0
When y = 1, x = \(\frac{1}{2}\)
∴ the points of intersection are 0(0, 0) and B(\(\frac{1}{2}\), 1)
Required area = area of the region OABCO = area of the region OABDO – area of the region OCBDO
Now, area of the region OABDO = area under the parabola y2 = 2x between x = 0 and x = \(\frac{1}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (i).1
Area of the region OCBDO = area under the line y = 2x between x = 0 and x = \(\frac{1}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (i).2

(ii) y2 = 4x and y = x.
Solution:
The vertex of the parabola y2 = 4x is at the origin O = (0, 0).
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (ii).jpg
To find the points of intersection of the line and the parabola, equaling the values of 4x from both the equations we get,
∴ y2 = y
∴ y2 – y = 0
∴ y(y – 1) = 0
∴ y = 0 or y = 1
When y = 0, x = \(\frac{0}{2}\) = 0
When y = 1, x = \(\frac{1}{2}\)
∴ the points of intersection are O(0, 0) and B(\(\frac{1}{2}\), 1)
Required area = area of the region OABCO = area of the region OABDO – area of the region OCBDO
Now, area of the region OABDO = area under the parabola y2 = 4x between x = 0 and x = \(\frac{1}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (ii).1
Area of the region OCBDO = area under the line y = 2x between x = 0 and x = \(\frac{1}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (ii).2

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1

(iii) y = x2 and the line y = 4x.
Solution:
The vertex of the parabola y = x2 is at the origin 0(0, 0)
To find the points of the intersection of a line and the parabola.
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (iii)
Equating the values of y from the two equations, we get
x2 = 4x
∴ x2 – 4x = 0
∴ x(x – 4) = 0
∴ x = 0, x = 4
When x = 0, y = 4(0) = 0
When x = 4, y = 4(4) = 16
∴ the points of intersection are 0(0, 0) and B(4, 16)
Required area = area of the region OABCO = (area of the region ODBCO) – (area of the region ODBAO)
Now, area of the region ODBCO = area under the line y = 4x between x = 0 and x = 4
= \(\int_{0}^{4} y d x\), where y = 4x
= \(\int_{0}^{4} 4 x d x\)
= 4\(\int_{0}^{4} x d x\)
= 4\([latex]\int_{0}^{4} x d x\)[/latex]
= 2(16 – 0)
= 32
Area of the region ODBAO = area under the parabola y = x2 between x = 0 and x = 4
= \(\int_{0}^{4} y d x\), where y = x2
= \(\int_{0}^{4} x^{2} d x\)
= \(\left[\frac{x^{3}}{3}\right]_{0}^{4}\)
= \(\frac{1}{3}\) (64 – 0)
= \(\frac{64}{3}\)
∴ required area = 32 – \(\frac{64}{3}\) = \(\frac{32}{3}\) sq units.

(iv) y2 = 4ax and y = x.
Solution:
The vertex of the parabola y2 = 4ax is at the origin O = (0, 0).
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (iv).jpg
To find the points of intersection of the line and the parabola, equaling the values of 4ax from both the equations we get,
∴ y2 = y
∴ y2 – y = 0
∴ y(y – 1) = 0
∴ y = 0 or y = 1
When y = 0, x = \(\frac{0}{2}\) = 0
When y = 1, x = \(\frac{1}{2}\)
∴ the points of intersection are O(0, 0) and B(\(\frac{1}{2}\), 1)
Required area = area of the region OABCO = area of the region OABDO – area of the region OCBDO
Now, area of the region OABDO
= area under the parabola y2 = 4ax between x = 0 and x = \(\frac{1}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (iv).1
Area of the region OCBDO
= area under the line y
= 4ax between x = 0 and x = \(\frac{1}{4 a x}\)
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (iv).2

Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1

(v) y = x2 + 3 and y = x + 3.
Solution:
The given parabola is y = x2 + 3, i.e. (x – 0)2 = y – 3
∴ its vertex is P(0, 3).
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (v)
To find the points of intersection of the line and the parabola.
Equating the values of y from both the equations, we get
x2 + 3 = x + 3
∴ x2 – x = 0
∴ x(x – 1) = 0
∴ x = 0 or x = 1
When x = 0, y = 0 + 3 = 3
When x = 1, y = 1 + 3 = 4
∴ the points of intersection are P(0, 3) and B(1, 4)
Required area = area of the region PABCP = area of the region OPABDO – area of the region OPCBDO
Now, area of the region OPABDO
= area under the line y = x + 3 between x = 0 and x = 1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (v).1
Area of the region OPCBDO = area under the parabola y = x2 + 3 between x = 0 and x = 1
Maharashtra Board 12th Maths Solutions Chapter 5 Application of Definite Integration Ex 5.1 Q3 (v).2

Class 12 Maharashtra State Board Maths Solution 

Definite Integration Class 12 Maths 2 Miscellaneous Exercise 4 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 4 Definite Integration Miscellaneous Exercise 4 Questions and Answers.

12th Maths Part 2 Definite Integration Miscellaneous Exercise 4 Questions And Answers Maharashtra Board

I. Choose the correct option from the given alternatives:

Question 1.
\(\int_{2}^{3} \frac{d x}{x\left(x^{3}-1\right)}=\)
(a) \(\frac{1}{3} \log \left(\frac{208}{189}\right)\)
(b) \(\frac{1}{3} \log \left(\frac{189}{208}\right)\)
(c) \(\log \left(\frac{208}{189}\right)\)
(d) \(\log \left(\frac{189}{208}\right)\)
Answer:
(a) \(\frac{1}{3} \log \left(\frac{208}{189}\right)\)

Question 2.
\(\int_{0}^{\pi / 2} \frac{\sin ^{2} x \cdot d x}{(1+\cos x)^{2}}=\)
(a) \(\frac{4-\pi}{2}\)
(b) \(\frac{\pi-4}{2}\)
(c) 4 – \(\frac{\pi}{2}\)
(d) \(\frac{4+\pi}{2}\)
Answer:
(a) \(\frac{4-\pi}{2}\)

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 3.
\(\int_{0}^{\log 5} \frac{e^{x} \sqrt{e^{x}-1}}{e^{x}+3} \cdot d x=\)
(a) 3 + 2π
(b) 4 – π
(c) 2 + π
(d) 4 + π
Answer:
(b) 4 – π

Question 4.
\(\int_{0}^{\pi / 2} \sin ^{6} x \cos ^{2} x \cdot d x=\)
(a) \(\frac{7 \pi}{256}\)
(b) \(\frac{3 \pi}{256}\)
(c) \(\frac{5 \pi}{256}\)
(d) \(\frac{-5 \pi}{256}\)
Answer:
(c) \(\frac{5 \pi}{256}\)

Question 5.
If \(\int_{0}^{1} \frac{d x}{\sqrt{1+x}-\sqrt{X}}=\frac{k}{3}\), then k is equal to
(a) √2(2√2 – 2)
(b) \(\frac{\sqrt{2}}{3}\)(2 – 2√2)
(c) \(\frac{2 \sqrt{2}-2}{3}\)
(d) 4√2
Answer:
(d) 4√2

Question 6.
\(\int_{1}^{2} \frac{1}{x^{2}} e^{\frac{1}{x}} \cdot d x=\)
(a) √e + 1
(b) √e − 1
(c) √e(√e − 1)
(d) \(\frac{\sqrt{e}-1}{e}\)
Answer:
(c) √e(√e − 1)

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 7.
If \(\int_{2}^{e}\left[\frac{1}{\log x}-\frac{1}{(\log x)^{2}}\right] \cdot d x=a+\frac{b}{\log 2}\), then
(a) a = e, b = -2
(b) a = e, b = 2
(c) a = -e, b = 2
(d) a = -e, b = -2
Answer:
(a) a = e, b = -2

Question 8.
Let \(\mathrm{I}_{1}=\int_{e}^{e^{2}} \frac{d x}{\log x}\) and \(\mathrm{I}_{2}=\int_{1}^{2} \frac{e^{x}}{\boldsymbol{X}} \cdot d x\), then
(a) I1 = \(\frac{1}{3}\) I2
(b) I1 + I2 = 0
(c) I1 = 2I2
(d) I1 = I2
Answer:
(d) I1 = I2

Question 9.
\(\int_{0}^{9} \frac{\sqrt{X}}{\sqrt{X}+\sqrt{9-X}} \cdot d x=\)
(a) 9
(b) \(\frac{9}{2}\)
(c) 0
(d) 1
Answer:
(b) \(\frac{9}{2}\)

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 10.
The value of \(\int_{-\pi / 4}^{\pi / 4} \log \left(\frac{2+\sin \theta}{2-\sin \theta}\right) \cdot d \theta\) is
(a) 0
(b) 1
(c) 2
(d) π
Answer:
(a) 0

II. Evaluate the following:

Question 1.
\(\int_{0}^{\pi / 2} \frac{\cos x}{3 \cos x+\sin x} d x\)
Solution:
Let I = \(\int_{0}^{\pi / 2} \frac{\cos x}{3 \cos x+\sin x} d x\)
Put Numerator = A(Denominator) + B[\(\frac{d}{d x}\)(Denominator)]
∴ cos x = A(3 cos x + sin x) + B[\(\frac{d}{d x}\)(3 cos x + sin x)]
= A(3 cos x + sin x) + B(-3 sin x + cos x)
∴ cos x + 0 . sin x = (3A + B) cos x + (A – 3B) sin x
Comparing the coefficients of sinx and cos x on both the sides, we get
3A + B = 1 ………. (1)
A – 3B = 0 ………. (2)
Multiplying equation (1) by 3, we get
9A + 3B = 3 ………(3)
Adding (2) and (3), we get
10A = 3
∴ A = \(\frac{3}{10}\)
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q1.1

Question 2.
\(\int_{\pi / 4}^{\pi / 2} \frac{\cos \theta}{\left[\cos \frac{\theta}{2}+\sin \frac{\theta}{2}\right]^{3}} d \theta\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q2
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q2.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 3.
\(\int_{0}^{1} \frac{1}{1+\sqrt{x}} d x\)
Solution:
Let I = \(\int_{0}^{1} \frac{1}{1+\sqrt{x}} d x\)
Put √x = t
∴ x = t2 and dx = 2t . dt
When x = 0, t = 0
When x = 1, t = 1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q3

Question 4.
\(\int_{0}^{\pi / 4} \frac{\tan ^{3} x}{1+\cos 2 x} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q4

Question 5.
\(\int_{0}^{1} t^{5} \sqrt{1-t^{2}} d t\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q5

Question 6.
\(\int_{0}^{1}\left(\cos ^{-1} x\right)^{2} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q6
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q6.1

Question 7.
\(\int_{-1}^{1} \frac{1+x^{3}}{9-x^{2}} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q7
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q7.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q7.2

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 8.
\(\int_{0}^{\pi} x \cdot \sin x \cdot \cos ^{4} x d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q8
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q8.1

Question 9.
\(\int_{0}^{\pi} \frac{x}{1+\sin ^{2} x} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q9
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q9.1

Question 10.
\(\int_{1}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q10
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 II Q10.1

III. Evaluate the following:

Question 1.
\(\int_{0}^{1}\left(\frac{1}{1+x^{2}}\right) \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 2.
\(\int_{0}^{\pi / 2} \frac{1}{6-\cos x} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q2
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q2.1

Question 3.
\(\int_{0}^{a} \frac{1}{a^{2}+a x-x^{2}} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q3
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q3.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q3.2

Question 4.
\(\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x}{\sin x+\cos x} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q4
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q4.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 5.
\(\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x\)
Solution:
Let I = \(\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x\)
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q5

Question 6.
\(\int_{0}^{\pi / 4} \frac{\cos 2 x}{1+\cos 2 x+\sin 2 x} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q6
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q6.1

Question 7.
\(\int_{0}^{\pi / 2}[2 \log (\sin x)-\log (\sin 2 x)] d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q7

Question 8.
\(\int_{0}^{\pi}\left(\sin ^{-1} x+\cos ^{-1} x\right)^{3} \sin ^{3} x d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q8
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q8.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 9.
\(\int_{0}^{4}\left[\sqrt{x^{2}+2 x+3}\right]^{-1} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q9

Question 10.
\(\int_{-2}^{3}|x-2| d x\)
Solution:
|x – 2|= 2 – x, if x < 2
= x – 2, if x ≥ 2
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 III Q10

IV. Evaluate the following:

Question 1.
If \(\int_{a}^{a} \sqrt{x} d x=2 a \int_{0}^{\pi / 2} \sin ^{3} x d x\), find the value of \(\int_{a}^{a+1} x d x\).
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 IV Q1

Question 2.
If \(\int_{0}^{k} \frac{1}{2+8 x^{2}} \cdot d x=\frac{\pi}{16}\), find k.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 IV Q2
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 IV Q2.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4

Question 3.
If f(x) = a + bx + cx2, show that \(\int_{0}^{1} f(x) d x=\frac{1}{6}\left[f(0)+4 f\left(\frac{1}{2}\right)+f(1)\right]\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 IV Q3
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Miscellaneous Exercise 4 IV Q3.1

Class 12 Maharashtra State Board Maths Solution 

Definite Integration Class 12 Maths 2 Exercise 4.2 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 4 Definite Integration Ex 4.2 Questions and Answers.

12th Maths Part 2 Definite Integration Exercise 4.2 Questions And Answers Maharashtra Board

I. Evaluate:

Question 1.
\(\int_{1}^{9} \frac{x+1}{\sqrt{x}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q1

Question 2.
\(\int_{2}^{3} \frac{1}{x^{2}+5 x+6} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q2

Question 3.
\(\int_{0}^{\pi / 4} \cot ^{2} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q3
The integral does not exist since cot 0 is not defined.

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 4.
\(\int_{-\pi / 4}^{\pi / 4} \frac{1}{1-\sin x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q4
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q4.1

Question 5.
\(\int_{3}^{5} \frac{1}{\sqrt{2 x+3}-\sqrt{2 x-3}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q5

Question 6.
\(\int_{0}^{1} \frac{x^{2}-2}{x^{2}+1} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q6
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q6.1

Question 7.
\(\int_{0}^{\pi / 4} \sin 4 x \sin 3 x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q7

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 8.
\(\int_{0}^{\pi / 4} \sqrt{1+\sin 2 x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q8

Question 9.
\(\int_{0}^{\pi / 4} \sin ^{4} x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q9
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q9.1

Question 10.
\(\int_{-4}^{2} \frac{1}{x^{2}+4 x+13} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q10

Question 11.
\(\int_{0}^{4} \frac{1}{\sqrt{4 x-x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q11

Question 12.
\(\int_{0}^{1} \frac{1}{\sqrt{3+2 x-x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q12

Question 13.
\(\int_{0}^{\pi / 2} x \cdot \sin x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q13
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q13.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 14.
\(\int_{0}^{1} x \cdot \tan ^{-1} x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q14

Question 15.
\(\int_{0}^{\infty} x \cdot e^{-x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q15
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 I Q15.1

II. Evaluate:

Question 1.
\(\int_{0}^{\frac{1}{\sqrt{2}}} \frac{\sin ^{-1} x}{\left(1-x^{2}\right)^{\frac{3}{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q1.1

Question 2.
\(\int_{0}^{\pi / 4} \frac{\sec ^{2} x}{3 \tan ^{2} x+4 \tan x+1} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q2

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 3.
\(\int_{0}^{4 \pi} \frac{\sin 2 x}{\sin ^{4} x+\cos ^{4} x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q3
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q3.1

Question 4.
\(\int_{0}^{2 \pi} \sqrt{\cos x} \cdot \sin ^{3} x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q4

Question 5.
\(\int_{0}^{\pi / 2} \frac{1}{5+4 \cos x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q5

Question 6.
\(\int_{0}^{\pi / 4} \frac{\cos x}{4-\sin ^{2} x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q6

Question 7.
\(\int_{0}^{\pi / 2} \frac{\cos X}{(1+\sin x)(2+\sin x)} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q7
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q7.1

Question 8.
\(\int_{-1}^{1} \frac{1}{a^{2} e^{x}+b^{2} e^{-x}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q8

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 9.
\(\int_{0}^{\pi} \frac{1}{3+2 \sin x+\cos x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q9
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q9.1

Question 10.
\(\int_{0}^{\pi / 4} \sec ^{4} x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q10

Question 11.
\(\int_{0}^{1} \sqrt{\frac{1-x}{1+x}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q11
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q11.1

Question 12.
\(\int_{0}^{\pi} \sin ^{3} x(1+2 \cos x)(1+\cos x)^{2} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q12
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q12.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q12.2

Question 13.
\(\int_{0}^{\pi / 2} \sin 2 x \cdot \tan ^{-1}(\sin x) \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q13
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q13.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 14.
\(\int_{\frac{1}{\sqrt{2}}}^{1} \frac{\left(e^{\cos ^{-1} x}\right)\left(\sin ^{-1} x\right)}{\sqrt{1-x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q14
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q14.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q14.2

Question 15.
\(\int_{2}^{3} \frac{\cos (\log x)}{x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 II Q15

III. Evaluate:

Question 1.
\(\int_{0}^{a} \frac{1}{x+\sqrt{a^{2}-x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q1.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 2.
\(\int_{0}^{\pi / 2} \log \tan x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q2
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q2.1

Question 3.
\(\int_{0}^{1} \log \left(\frac{1}{x}-1\right) \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q3

Question 4.
\(\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cdot \cos x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q4
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q4.1

Question 5.
\(\int_{0}^{3} x^{2}(3-x)^{\frac{5}{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q5

Question 6.
\(\int_{-3}^{3} \frac{x^{3}}{9-x^{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q6

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 7.
\(\int_{-\pi / 2}^{\pi / 2} \log \left(\frac{2+\sin x}{2-\sin x}\right) \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q7
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q7.1

Question 8.
\(\int_{-\pi / 4}^{\pi / 4} \frac{x+\frac{\pi}{4}}{2-\cos 2 x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q8
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q8.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q8.2

Question 9.
\(\int_{-\pi / 4}^{\pi / 4} x^{3} \cdot \sin ^{4} x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q9

Question 10.
\(\int_{0}^{1} \frac{\log (x+1)}{x^{2}+1} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q10
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q10.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 11.
\(\int_{-1}^{1} \frac{x^{3}+2}{\sqrt{x^{2}+4}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q11
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q11.1

Question 12.
\(\int_{-a}^{a} \frac{x+x^{3}}{16-x^{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q12

Question 13.
\(\int_{0}^{1} t^{2} \sqrt{1-t} \cdot d t\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q13

Question 14.
\(\int_{0}^{\pi} x \cdot \sin x \cdot \cos ^{2} x \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q14
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q14.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q14.2

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2

Question 15.
\(\int_{0}^{1} \frac{\log x}{\sqrt{1-x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q15
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q15.1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q15.2
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q15.3
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.2 III Q15.4

Class 12 Maharashtra State Board Maths Solution 

Definite Integration Class 12 Maths 2 Exercise 4.1 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 4 Definite Integration Ex 4.1 Questions and Answers.

12th Maths Part 2 Definite Integration Exercise 4.1 Questions And Answers Maharashtra Board

I. Evaluate the following integrals as a limit of a sum.

Question 1.
\(\int_{1}^{3}(3 x-4) \cdot d x\)
Solution:
Let f(x) = 3x – 4, for 1 ≤ x ≤ 3
Divide the closed interval [1, 3] into n subintervals each of length h at the points
1, 1 + h, 1 + 2h, 1 + rh, ….., 1 + nh = 3
∴ nh = 2
∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0
Here, a = 1
∴ f(a + rh) = f(1 + rh)
= 3(1 + rh) – 4
= 3rh – 1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q1.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1

Question 2.
\(\int_{0}^{4} x^{2} d x\)
Solution:
Let f(x) = x2, for 0 ≤ x ≤ 4
Divide the closed interval [0, 4] into n subintervals each of length h at the points
0, 0 + h, 0 + 2h, ….., 0 + rh, ….., 0 + nh = 4
i.e. 0, h, 2h, ….., rh, ….., nh = 4
∴ h = \(\frac{4}{n}\) as n → ∞, h → 0
Here, a = 0
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q2

Question 3.
\(\int_{0}^{2} e^{x} d x\)
Solution:
Let f(x) = ex, for 0 ≤ x ≤ 2
Divide the closed interval [0, 2] into n equal subntervals each of length h at the points
0, 0 + h, 0 + 2h, ….., 0 + rh, ….., 0 + nh = 2
i.e. 0, h, 2h, ….., rh, ….., nh = 2
∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0
Here, a = 0
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q3
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q3.1

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1

Question 4.
\(\int_{0}^{2}\left(3 x^{2}-1\right) d x\)
Solution:
Let f(x) = 3x2 – 1, for 0 ≤ x ≤ 2
Divide the closed interval [0, 2] into n subintervals each of length h at the points.
0, 0 + h, 0 + 2h, ….., 0 + rh, ……, 0 + nh = 2
i.e. 0, h, 2h, ….., rh, ….., nh = 2
∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0
Here, a = 0
∴ f(a + rh) = f(0 + rh)
= f(rh)
= 3(rh)2 – 1
= 3r2h2 – 1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q4

Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1

Question 5.
\(\int_{1}^{3} x^{3} d x\)
Solution:
Let f(x) = x3, for 1 ≤ x ≤ 3.
Divide the closed interval [1, 3] into n equal su bintervals each of length h at the points
1, 1 + h, 1 + 2h, ……, 1 + rh, ……, 1 + nh = 3
∴ nh = 2
∴ h = \(\frac{2}{n}\) and as n → ∞, h → 0
Here a = 1
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q5
Maharashtra Board 12th Maths Solutions Chapter 4 Definite Integration Ex 4.1 Q5.1

Class 12 Maharashtra State Board Maths Solution 

Indefinite Integration Class 12 Maths 2 Miscellaneous Exercise 3 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Miscellaneous Exercise 3 Questions and Answers.

12th Maths Part 2 Indefinite Integration Miscellaneous Exercise 3 Questions And Answers Maharashtra Board

I. Choose the correct options from the given alternatives:

Question 1.
\(\int \frac{1+x+\sqrt{x+x^{2}}}{\sqrt{x}+\sqrt{1+x}} \cdot d x=\)
(a) \(\frac{1}{2} \sqrt{x+1}+c\)
(b) \(\frac{2}{3}(x+1)^{\frac{3}{2}}+c\)
(c) \(\sqrt{x+1}+c\)
(d) \(2(x+1)^{\frac{3}{2}}+c\)
Answer:
(b) \(\frac{2}{3}(x+1)^{\frac{3}{2}}+c\)

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 2.
\(\int \frac{1}{x+x^{5}} \cdot d x\) = f(x) + c, then \(\int \frac{x^{4}}{x+x^{5}} \cdot d x=\)
(a) log x – f(x) + c
(b) f(x) + log x + c
(c) f(x) – log x + c
(d) \(\frac{1}{5}\) x5 f(x) + c
Answer:
(a) log x – f(x) + c
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q2

Question 3.
\(\int \frac{\log (3 x)}{x \log (9 x)} \cdot d x=\)
(a) log(3x) – log(9x) + c
(b) log(x) – (log 3) . log(log 9x) + c
(c) log 9 – (log x) . log(log 3x) + c
(d) log(x) + log(3) . log(log 9x) + c
Answer:
(b) log(x) – (log 3) . log(log 9x) + c
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q3

Question 4.
\(\int \frac{\sin ^{m} X}{\cos ^{m+2} X} \cdot d x=\)
(a) \(\frac{\tan ^{m+1} \boldsymbol{X}}{m+1}+c\)
(b) (m + 2) tanm+1 x + c
(c) \(\frac{\tan ^{m} \boldsymbol{X}}{m}+c\)
(d) (m + 1) tanm+1 x + c
Answer:
(a) \(\frac{\tan ^{m+1} \boldsymbol{X}}{m+1}+c\)

Question 5.
∫tan(sin-1 x) . dx =
(a) \(\left(1-x^{2}\right)^{-\frac{1}{2}}+c\)
(b) \(\left(1-x^{2}\right)^{\frac{1}{2}}+c\)
(c) \(\frac{\tan ^{m} \boldsymbol{X}}{\sqrt{1-x^{2}}}+c\)
(d) \(-\sqrt{1-x^{2}}+c\)
Answer:
(d) \(-\sqrt{1-x^{2}}+c\)

Hint: sin-1 x = \(\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right)\)

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 6.
\(\int \frac{x-\sin x}{1-\cos x} \cdot d x=\)
(a) x cot(\(\frac{x}{2}\)) + c
(b) -x cot(\(\frac{x}{2}\)) + c
(c) cot(\(\frac{x}{2}\)) + c
(d) x tan(\(\frac{x}{2}\)) + c
Answer:
(b) -x cot(\(\frac{x}{2}\)) + c
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q6
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q6.1

Question 7.
If f(x) = \(\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}\), g(x) = \(e^{\sin ^{-1} x}\), then ∫f(x) . g(x) . dx =
(a) \(e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x-1\right)+c\)
(b) \(e^{\sin ^{-1} x} \cdot\left(1-\sin ^{-1} x\right)+c\)
(c) \(e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x+1\right)+c\)
(d) \(e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} X-1\right)+c\)
Answer:
(a) \(e^{\sin ^{-1} x} \cdot\left(\sin ^{-1} x-1\right)+c\)

Question 8.
If ∫tan3 x . sec3 x . dx = (\(\frac{1}{m}\)) secm x – (\(\frac{1}{n}\)) secn x + c, then (m, n) =
(a) (5, 3)
(b) (3, 5)
(c) \(\left(\frac{1}{5}, \frac{1}{3}\right)\)
(d) (4, 4)
Answer:
(a) (5, 3)

Hint: ∫tan3 x . sec3 x dx
= ∫sec2 x . tan2 x . sec x tan x dx
= ∫sec2 x (sec2 x – 1) sec x tan x dx
Put sec x = t.

Question 9.
\(\int \frac{1}{\cos x-\cos ^{2} x} \cdot d x=\)
(a) log(cosec x – cot x) + tan(\(\frac{x}{2}\)) + c
(b) sin 2x – cos x + c
(c) log(sec x + tan x) – cot(\(\frac{x}{2}\)) + c
(d) cos 2x – sin x + c
Answer:
(c) log(sec x + tan x) – cot(\(\frac{x}{2}\)) + c
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q9

Question 10.
\(\int \frac{\sqrt{\cot x}}{\sin x \cdot \cos x} \cdot d x=\)
(a) \(2 \sqrt{\cot x}+c\)
(b) \(-2 \sqrt{\cot x}+c\)
(c) \(\frac{1}{2} \sqrt{\cot x}+c\)
(d) \(\sqrt{\cot X}+c\)
Answer:
(b) \(-2 \sqrt{\cot x}+c\)

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 11.
\(\int \frac{e^{x}(x-1)}{x^{2}} \cdot d x=\)
(a) \(\frac{e^{x}}{x}+c\)
(b) \(\frac{e^{x}}{x^{2}}+c\)
(c) \(\left(x-\frac{1}{x}\right) e^{x}+c\)
(d) x e-x + c
Answer:
(a) \(\frac{e^{x}}{x}+c\)

Question 12.
∫sin(log x) . dx =
(a) \(\frac{x}{2}\) [sin(log x) – cos(log x)] + c
(b) \(\frac{x}{2}\) [sin(log x) + cos(log x)] + c
(c) \(\frac{x}{2}\) [cos(log x) – sin(log x)] + c
(d) \(\frac{x}{4}\) [cos(log x) – sin(log x)] + c
Answer:
(a) \(\frac{x}{2}\) [sin(log x) – cos(log x)] + c

Question 13.
∫xx (1 + log x) . dx =
(a) \(\frac{1}{2}\) (1 + log x)2 + c
(b) x2x + c
(c) xx log x + c
(d) xx + c
Answer:
(d) xx + c

Hint: \(\frac{d}{d x}\)(xx) = xx (1 + log x)

Question 14.
\(\int \cos ^{-\frac{3}{7}} x \cdot \sin ^{-\frac{11}{7}} x \cdot d x=\)
(a) \(\log \left(\sin ^{-\frac{4}{7}} x\right)+c\)
(b) \(\frac{4}{7} \tan ^{\frac{4}{7}} x+c\)
(c) \(-\frac{7}{4} \tan ^{-\frac{4}{7}} x+c\)
(d) \(\log \left(\cos ^{\frac{3}{7}} x\right)+c\)
Answer:
(c) \(-\frac{7}{4} \tan ^{-\frac{4}{7}} x+c\)

Hint: \(\int \cos ^{-\frac{3}{7}} x \sin ^{-\frac{11}{7}} x d x\)
= \(\int \frac{\sin ^{-\frac{11}{7}} x}{\cos ^{-\frac{11}{7}} x \cdot \cos ^{2} x} d x\)
= \(\int \tan ^{-\frac{11}{7}} x \sec ^{2} x d x\)
Put tan x = t.

Question 15.
\(2 \int \frac{\cos ^{2} x-\sin ^{2} x}{\cos ^{2} x+\sin ^{2} x} \cdot d x=\)
(a) sin 2x + c
(b) cos 2x + c
(c) tan 2x + c
(d) 2 sin 2x + c
Answer:
(a) sin 2x + c

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 16.
\(\int \frac{d x}{\cos x \sqrt{\sin ^{2} x-\cos ^{2} x}} \cdot d x=\)
(a) log(tan x – \(\sqrt{\tan ^{2} x-1}\)) + c
(b) sin-1 (tan x) + c
(c) 1 + sin-1 (cot x) + c
(d) log(tan x + \(\sqrt{\tan ^{2} x-1}\)) + c
Answer:
(d) log(tan x + \(\sqrt{\tan ^{2} x-1}\)) + c
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q16

Question 17.
\(\int \frac{\log x}{(\log e x)^{2}} \cdot d x=\)
(a) \(\frac{x}{1+\log x}+c\)
(b) x(1 + log x) + c
(c) \(\frac{1}{1+\log x}+c\)
(d) \(\frac{1}{1-\log x}+c\)
Answer:
(a) \(\frac{x}{1+\log x}+c\)

Question 18.
∫[sin(log x) + cos(log x)] . dx =
(a) x cos(log x) + c
(b) sin(log x) + c
(c) cos(log x) + c
(d) x sin(log x) + c
Answer:
(d) x sin(log x) + c

Question 19.
\(\int \frac{\cos 2 x-1}{\cos 2 x+1} \cdot d x=\)
(a) tan x – x + c
(b) x + tan x + c
(c) x – tan x + c
(d) -x – cot x + c
Answer:
(c) x – tan x + c
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q19

Question 20.
\(\int \frac{e^{2 x}+e^{-2 x}}{e^{x}} \cdot d x=\)
(a) \(e^{x}-\frac{1}{3 e^{3 x}}+c\)
(b) \(e^{x}+\frac{1}{3 e^{3 x}}+c\)
(c) \(e^{-x}+\frac{1}{3 e^{3 x}}+c\)
(d) \(e^{-x}-\frac{1}{3 e^{3 x}}+c\)
Answer:
(a) \(e^{x}-\frac{1}{3 e^{3 x}}+c\)
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 I Q20

II. Integrate the following with respect to the respective variable:

Question 1.
(x – 2)2 √x
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q1.1

Question 2.
\(\frac{x^{7}}{x+1}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q2

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 3.
\((6 x+5)^{\frac{3}{2}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q3

Question 4.
\(\frac{t^{3}}{(t+1)^{2}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q4
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q4.1

Question 5.
\(\frac{3-2 \sin x}{\cos ^{2} x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q5

Question 6.
\(\frac{\sin ^{6} \theta+\cos ^{6} \theta}{\sin ^{2} \theta \cdot \cos ^{2} \theta}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q6

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 7.
cos 3x cos 2x cos x
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q7
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q7.1

Question 8.
\(\frac{\cos 7 x-\cos 8 x}{1+2 \cos 5 x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q8
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q8.1

Question 9.
\(\cot ^{-1}\left(\frac{1+\sin x}{\cos x}\right)\)
Solution:
Let I = \(\int \cot ^{-1}\left(\frac{1+\sin x}{\cos x}\right) d x\)
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 II Q9

III. Integrate the following w.r.t. x:

Question 1.
\(\frac{(1+\log x)^{3}}{x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 2.
cot-1 (1 – x + x2)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q2.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q2.2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q2.3
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q2.4

Question 3.
\(\frac{1}{x \sin ^{2}(\log x)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q3

Question 4.
\(\sqrt{x} \sec \left(x^{\frac{3}{2}}\right) \tan \left(x^{\frac{3}{2}}\right)\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q4

Question 5.
log(1 + cos x) – x tan(\(\frac{x}{2}\))
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q5
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q5.1

Question 6.
\(\frac{x^{2}}{\sqrt{1-x^{6}}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q6

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 7.
\(\frac{1}{(1-\cos 4 x)(3-\cot 2 x)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q7

Question 8.
log(log x) + (log x)-2
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q8
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q8.1

Question 9.
\(\frac{1}{2 \cos x+3 \sin x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q9
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q9.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q9.2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q9.3

Question 10.
\(\frac{1}{x^{3} \sqrt{x^{2}-1}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q10
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q10.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 11.
\(\frac{3 x+1}{\sqrt{-2 x^{2}+x+3}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q11
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q11.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q11.2

Question 12.
log(x2 + 1)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q12

Question 13.
e2x sin x cos x
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q13
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q13.1

Question 14.
\(\frac{x^{2}}{(x-1)(3 x-1)(3 x-2)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q14
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q14.1

Question 15.
\(\frac{1}{\sin x+\sin 2 x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q15
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q15.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q15.2

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 16.
\(\sec ^{2} x \sqrt{7+2 \tan x-\tan ^{2} x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q16
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q16.1

Question 17.
\(\frac{x+5}{x^{3}+3 x^{2}-x-3}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q17
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q17.1

Question 18.
\(\frac{1}{x\left(x^{5}+1\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q18
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q18.1

Question 19.
\(\frac{\sqrt{\tan x}}{\sin x \cdot \cos x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q19

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3

Question 20.
sec4 x cosec2 x
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Miscellaneous Exercise 3 III Q20

Class 12 Maharashtra State Board Maths Solution 

Indefinite Integration Class 12 Maths 2 Exercise 3.4 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Ex 3.4 Questions and Answers.

12th Maths Part 2 Indefinite Integration Exercise 3.4 Questions And Answers Maharashtra Board

I. Integrate the following w. r. t. x:

Question 1.
\(\frac{x^{2}+2}{(x-1)(x+2)(x+3)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q1.1

Question 2.
\(\frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}-2\right)\left(x^{2}+3\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q2.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q2.2

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4

Question 3.
\(\frac{12 x+3}{6 x^{2}+13 x-63}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q3
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q3.1

Question 4.
\(\frac{2 x}{4-3 x-x^{2}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q4

Question 5.
\(\frac{x^{2}+x-1}{x^{2}+x-6}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q5
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q5.1

Question 6.
\(\frac{6 x^{3}+5 x^{2}-7}{3 x^{2}-2 x-1}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q6
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q6.1

Question 7.
\(\frac{12 x^{2}-2 x-9}{\left(4 x^{2}-1\right)(x+3)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q7
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q7.1

Question 8.
\(\frac{1}{x\left(x^{5}+1\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q8

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4

Question 9.
\(\frac{2 x^{2}-1}{x^{4}+9 x^{2}+20}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q9
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q9.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q9.2

Question 10.
\(\frac{x^{2}+3}{\left(x^{2}-1\right)\left(x^{2}-2\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q10
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q10.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q10.2

Question 11.
\(\frac{2 x}{\left(2+x^{2}\right)\left(3+x^{2}\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q11

Question 12.
\(\frac{2^{x}}{4^{x}-3 \cdot 2^{x}-4}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q12
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q12.1

Question 13.
\(\frac{3 x-2}{(x+1)^{2}(x+3)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q13
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q13.1

Question 14.
\(\frac{5 x^{2}+20 x+6}{x^{3}+2 x^{2}+x}\)
Solution:
Let I = ∫\(\frac{5 x^{2}+20 x+6}{x^{3}+2 x^{2}+x}\) dx
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q14
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q14.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4

Question 15.
\(\frac{1}{x\left(1+4 x^{3}+3 x^{6}\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q15
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q15.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q15.2

Question 16.
\(\frac{1}{x^{3}-1}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q16
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q16.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q16.2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q16.3

Question 17.
\(\frac{(3 \sin x-2) \cdot \cos x}{5-4 \sin x-\cos ^{2} x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q17
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q17.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q17.2

Question 18.
\(\frac{1}{\sin x+\sin 2 x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q18
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q18.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q18.2

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4

Question 19.
\(\frac{1}{2 \sin x+\sin 2 x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q19
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q19.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q19.2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q19.3

Question 20.
\(\frac{1}{\sin 2 x+\cos x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q20
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q20.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q20.2

Question 21.
\(\frac{1}{\sin x \cdot(3+2 \cos x)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q21
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q21.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q21.2

Question 22.
\(\frac{5 \cdot e^{x}}{\left(e^{x}+1\right)\left(e^{2 x}+9\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q22
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q22.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4

Question 23.
\(\frac{2 \log x+3}{x(3 \log x+2)\left[(\log x)^{2}+1\right]}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q23
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q23.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.4 Q23.2

Class 12 Maharashtra State Board Maths Solution 

Indefinite Integration Class 12 Maths 2 Exercise 3.3 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Ex 3.3 Questions and Answers.

12th Maths Part 2 Indefinite Integration Exercise 3.3 Questions And Answers Maharashtra Board

I. Evaluate the following:

Question 1.
∫x2 log x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q1

Question 2.
∫x2 sin 3x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q2.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 3.
∫x tan-1 x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q3

Question 4.
∫x2 tan-1 x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q4

Question 5.
∫x3 tan-1 x dx
Solution:
Let I = ∫x3 tan-1 x dx
= ∫(tan-1 x) . x3 dx
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q5

Question 6.
∫(log x)2 dx
Solution:
Let I = ∫(log x)2 dx
Put log x = t
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q6
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q6.1

Question 7.
∫sec3 x dx
Solution:
Let I = ∫sec3 x dx
= ∫sec x sec2 x dx
= sec x ∫sec2 x dx – ∫[\(\frac{d}{d x}\)(sec x) ∫sec2 x dx] dx
= sec x tan x – ∫(sec x tan x)(tan x) dx
= sec x tan x – ∫sec x tan2 x dx
= sec x tan x – ∫sec x (sec2 x – 1) dx
= sec x tan x – ∫sec3 x dx + ∫sec x dx
∴ I = sec x tan x – I + log|sec x + tan x|
∴ 2I = sec x tan x + log|sec x + tan x|
∴ I = \(\frac{1}{2}\) [sec x tan x + log|sec x + tan x|] + c.

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 8.
∫x . sin2 x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q8

Question 9.
∫x3 log x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q9

Question 10.
∫e2x cos 3x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q10
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q10.1

Question 11.
∫x sin-1 x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q11
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q11.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 12.
∫x2 cos-1 x dx
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q12

Question 13.
\(\int \frac{\log (\log x)}{x} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q13
= t(log t – 1) + c
= (log x) . [log(log x) – 1] + c.

Question 14.
\(\int \frac{t \cdot \sin ^{-1} t}{\sqrt{1-t^{2}}} d t\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q14

Question 15.
∫cos√x dx
Solution:
Let I = ∫cos√x dx
Put √x = t
∴ x = t2
∴ dx = 2t dt
∴ I = ∫(cos t) 2t dt
= ∫2t cos t dt
= 2t ∫cos t dt – ∫[\(\frac{d}{d t}\)(2t) ∫cos t dt]dt
= 2t sin t – ∫2 sin t dt
= 2t sin t + 2 cos t + c
= 2[√x sin√x + cos√x] + c.

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 16.
∫sin θ . log(cos θ) dθ
Solution:
Let I = ∫sin θ . log (cos θ) dθ
= ∫log(cos θ) . sin θ dθ
Put cos θ = t
∴ -sin θ dθ = dt
∴ sin θ dθ = -dt
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q16
= -t log t + t + c
= -cos θ . log(cos θ) + cos θ + c
= -cos θ [log(cos θ) – 1] + c.

Question 17.
∫x cos3 x dx
Solution:
cos 3x = 4 cos3 x – 3 cos x
∴ cos3 x + 3 cos x = 4cos3x
∴ cos3 x = \(\frac{1}{4}\) cos 3x + \(\frac{3}{4}\) cos x
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q17

Question 18.
\(\int \frac{\sin (\log x)^{2}}{x} \cdot \log x d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q18

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 19.
\(\int \frac{\log x}{x} d x\)
Solution:
Let I = \(\int \frac{\log x}{x} d x\)
Put log x = t
\(\frac{1}{x}\) dx = dt
∴ I = ∫t dt
= \(\frac{1}{2}\) t2 + c
= \(\frac{1}{2}\) (log x)2 + c

Question 20.
∫x sin 2x cos 5x dx.
Solution:
Let I = ∫x sin 2x cos 5x dx
sin 2x cos 5x = \(\frac{1}{2}\)[2 sin 2x cos 5x]
= \(\frac{1}{2}\) [sin(2x + 5x) + sin(2x – 5x)]
= \(\frac{1}{2}\) [sin 7x – sin 3x]
∴ ∫sin 2x cos 5x dx = \(\frac{1}{2}\) [∫sin 7x dx – ∫sin 3x dx]
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q20

Question 21.
\(\int \cos (\sqrt[3]{x}) d x\)
Solution:
Let I = \(\int \cos (\sqrt[3]{x}) d x\)
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 I Q21

II. Integrate the following functions w.r.t. x:

Question 1.
e2x sin 3x
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q1.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 2.
e-x cos 2x
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q2.1

Question 3.
sin(log x)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q3

Question 4.
\(\sqrt{5 x^{2}+3}\)
Solution:
Let I = \(\sqrt{5 x^{2}+3}\) dx
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q4

Question 5.
\(x^{2} \sqrt{a^{2}-x^{6}}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q5

Question 6.
\(\sqrt{(x-3)(7-x)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q6

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 7.
\(\sqrt{4^{x}\left(4^{x}+4\right)}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q7

Question 8.
(x + 1) \(\sqrt{2 x^{2}+3}\)
Solution:
Let I = ∫(x + 1) \(\sqrt{2 x^{2}+3}\) dx
Let x + 1 = A[\(\frac{d}{d x}\)(2x2 + 3)] + B
= A(4x) + B
= 4Ax + B
Comparing the coefficients of x and constant term on both the sides, we get
4A = 1, B = 1
∴ A = \(\frac{1}{4}\), B = 1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q8
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q8.1

Question 9.
\(x \sqrt{5-4 x-x^{2}}\)
Solution:
Let I = ∫\(x \sqrt{5-4 x-x^{2}}\) dx
Let x = A[\(\frac{d}{d x}\)(5 – 4x – x2)] + B
= A[-4 – 2x] + B
= -2Ax + (B – 4A)
Comparing the coefficients of x and the constant term on both sides, we get
-2A = 1, B – 4A = 0
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q9
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q9.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 10.
\(\sec ^{2} x \sqrt{\tan ^{2} x+\tan x-7}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q10
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q10.1

Question 11.
\(\sqrt{x^{2}+2 x+5}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q11

Question 12.
\(\sqrt{2 x^{2}+3 x+4}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q12
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 II Q12.1

III. Integrate the following functions w.r.t. x:

Question 1.
[2 + cot x – cosec2 x] ex
Solution:
Let I = ∫ex [2 + cot x – cosec2 x] dx
Put f(x) = 2 + cot x
∴ f'(x) = \(\frac{d}{d x}\)(2 + cot x)
= \(\frac{d}{d x}\)(2) + \(\frac{d}{d x}\)(cot x)
= 0 – cosec2 x
= -cosec2 x
∴ I = ∫ex [f(x) + f'(x)] dx
= ex f(x) + c
= ex (2 + cot x) + c.

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 2.
\(\left(\frac{1+\sin x}{1+\cos x}\right) e^{x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 III Q2

Question 3.
\(e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\)
Solution:
Let I = ∫\(e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)\)
Let f(x) = \(\frac{1}{x}\), f'(x) = \(-\frac{1}{x^{2}}\)
∴ I = ∫ex [f(x) + f'(x)] dx
= ex f(x) + c
= ex . \(\frac{1}{x}\) + c

Question 4.
\(\left[\frac{x}{(x+1)^{2}}\right] e^{x}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 III Q4

Question 5.
\(\frac{e^{x}}{x}\) . [x(log x)2 + 2 log x]
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 III Q5

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 6.
\(e^{5 x}\left[\frac{5 x \log x+1}{x}\right]\)
Solution:
Let I = ∫\(e^{5 x}\left[\frac{5 x \log x+1}{x}\right]\)
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 III Q6

Question 7.
\(e^{\sin ^{-1} x}\left[\frac{x+\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}\right]\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 III Q7

Question 8.
log(1 + x)(1+x)
Solution :
Let I = ∫log(1 + x)(1+x) dx
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3 III Q8

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.3

Question 9.
cosec (log x)[1 – cot(log x)]
Solution:
Let I = ∫cosec (log x)[1 – cot(log x)] dx
Put log x = t
x = et
dx = et dt
I = ∫cosec t (1 – cot t). et dt
= ∫et [cosec t – cosec t cot t] dt
= ∫et [cosec t + \(\frac{d}{d t}\) (cosec t)] dt
= et cosec t + c ….. [∵ et [f(t) +f'(t) ] dt = et f(t) + c ]
= x . cosec(log x) + c.

Class 12 Maharashtra State Board Maths Solution 

Indefinite Integration Class 12 Maths 2 Exercise 3.2(C) Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Ex 3.2(C) Questions and Answers.

12th Maths Part 2 Indefinite Integration Exercise 3.2(C) Questions And Answers Maharashtra Board

I. Evaluate:

Question 1.
\(\int \frac{3 x+4}{x^{2}+6 x+5} d x\)
Solution:
Let I = \(\int \frac{3 x+4}{x^{2}+6 x+5} d x\)
Let 3x + 4 = A[\(\frac{d}{d x}\)(x2 + 6x + 5)] + B
= A(2x + B) + B
∴ 3x + 4 = 2Ax + (6A + B)
Comparing the coefficient of x and constant on both sides, we get
2A = 3 and 6A + B = 4
∴ A = \(\frac{3}{2}\) and 6(\(\frac{3}{2}\)) + B = 4
∴ B = -5
3x + 4 = \(\frac{3}{2}\) (2x + 6) – 5
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q1

Question 2.
\(\int \frac{2 x+1}{x^{2}+4 x-5} d x\)
Solution:
Let I = \(\int \frac{2 x+1}{x^{2}+4 x-5} d x\)
Let 2x + 1 = A[\(\frac{d}{d x}\)(x2 + 4x – 5)] + B
2x + 1 = A(2x + 1) + B
∴ 2x + 1 = 2Ax + (4A + B)
Comparing the coefficient of x and constant on both sides, we get
4A = 2 and 4A + B = 4
∴ A = \(\frac{3}{2}\) and 6(\(\frac{3}{2}\)) + B = 4
∴ B = -5
∴ 2x + 1 = \(\frac{3}{2}\)(2x + 1) – 5
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q2.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C)

Question 3.
\(\int \frac{2 x+3}{2 x^{2}+3 x-1} d x\)
Solution:
Let I = \(\int \frac{2 x+3}{2 x^{2}+3 x-1} d x\)
Let 2x+ 3 = A[\(\frac{d}{d x}\)(2x2 + 3x – 1)] + B
2x + 1 = A(4x + 3) + B
∴ 2x + 1 = 4Ax + (3A + B)
Comparing the coefficient of x and constant on both sides, we get
4A = 2 and 3A + B = 3
∴ A = \(\frac{1}{2}\) and 3(\(\frac{1}{2}\)) + B = 3
∴ B = \(\frac{3}{2}\)
∴ 2x + 3 = \(\frac{1}{2}\)(4x + 3) + \(\frac{3}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q3
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q3.1

Question 4.
\(\int \frac{3 x+4}{\sqrt{2 x^{2}+2 x+1}} d x\)
Solution:
Let I = \(\int \frac{3 x+4}{\sqrt{2 x^{2}+2 x+1}} d x\)
Let 3x + 4 = A[\(\frac{d}{d x}\)(2x2 + 2x + 1)] + B
∴ 3x + 4 = A (4x + 2) + B
∴ 3x + 4 = 4Ax + (2A + B)
Comparing the coefficient of x and the constant on both the sides, we get
4A = 3 and 2A + B = 4
∴ A = \(\frac{3}{4}\) and 2(\(\frac{3}{4}\)) + B = 4
∴ B = \(\frac{5}{2}\)
∴ 3x + 4 = \(\frac{3}{4}\) (4x + 2) + \(\frac{5}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q4
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q4.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C)

Question 5.
\(\int \frac{7 x+3}{\sqrt{3+2 x-x^{2}}} d x\)
Solution:
Let I = \(\int \frac{7 x+3}{\sqrt{3+2 x-x^{2}}} d x\)
Let 7x + 3 = A[\(\frac{d}{d x}\)(3 + 2x – x2)] + B
7x + 3 = A(2 – 2x) + B
∴ 7x + 3 = -2Ax + (2A + B)
Comparing the coefficient of x and constant on both the sides, we get
-2A = 7 and 2A + B = 3
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q5
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q5.1

Question 6.
\(\int \sqrt{\frac{x-7}{x-9}} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q6
Comparing the coefficients of x and constant term on both sides, we get
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q6.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q6.2

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C)

Question 7.
\(\int \sqrt{\frac{9-x}{x}} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q7
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q7.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q7.2

Question 8.
\(\int \frac{3 \cos x}{4 \sin ^{2} x+4 \sin x-1} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q8
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q8.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C)

Question 9.
\(\int \sqrt{\frac{e^{3 x}-e^{2 x}}{e^{x}+1}} d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q9
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q9.1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(C) Q9.2

Class 12 Maharashtra State Board Maths Solution 

Indefinite Integration Class 12 Maths 2 Exercise 3.2(B) Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Indefinite Integration Ex 3.2(B) Questions and Answers.

12th Maths Part 2 Indefinite Integration Exercise 3.2(B) Questions And Answers Maharashtra Board

I. Evaluate the following:

Question 1.
\(\int \frac{1}{4 x^{2}-3} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q1.1

Question 2.
\(\int \frac{1}{25-9 x^{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q2

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 3.
\(\int \frac{1}{7+2 x^{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q3

Question 4.
\(\int \frac{1}{\sqrt{3 x^{2}+8}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q4
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q4.1

Question 5.
\(\int \frac{1}{\sqrt{11-4 x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q5

Question 6.
\(\int \frac{1}{\sqrt{2 x^{2}-5}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q6
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q6.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 7.
\(\int \sqrt{\frac{9+x}{9-x}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q7

Question 8.
\(\int \sqrt{\frac{2+x}{2-x}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q8
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q8.1

Question 9.
\(\int \sqrt{\frac{10+x}{10-x}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q9
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q9.1

Question 10.
\(\int \frac{1}{x^{2}+8 x+12} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q10

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 11.
\(\int \frac{1}{1+x-x^{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q11

Question 12.
\(\int \frac{1}{4 x^{2}-20 x+17} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q12
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q12.1

Question 13.
\(\int \frac{1}{5-4 x-3 x^{2}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q13
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q13.1

Question 14.
\(\int \frac{1}{\sqrt{3 x^{2}+5 x+7}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q14
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q14.1

Question 15.
\(\int \frac{1}{\sqrt{x^{2}+8 x-20}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q15

Question 16.
\(\int \frac{1}{\sqrt{8-3 x+2 x^{2}}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q16

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 17.
\(\int \frac{1}{\sqrt{(x-3)(x+2)}} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q17
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q17.1

Question 18.
\(\int \frac{1}{4+3 \cos ^{2} x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q18

Question 19.
\(\int \frac{1}{\cos 2 x+3 \sin ^{2} x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q19
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q19.1

Question 20.
\(\int \frac{\sin x}{\sin 3 x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) I Q20

II. Integrate the following functions w. r. t. x:

Question 1.
\(\int \frac{1}{3+2 \sin x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q1
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q1.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 2.
\(\int \frac{1}{4-5 \cos x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q2
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q2.1

Question 3.
\(\int \frac{1}{2+\cos x-\sin x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q3
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q3.1

Question 4.
\(\int \frac{1}{3+2 \sin x-\cos x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q4

Question 5.
\(\int \frac{1}{3-2 \cos 2 x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q5
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q5.1

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 6.
\(\int \frac{1}{2 \sin 2 x-3} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q6
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q6.1

Question 7.
\(\int \frac{1}{3+2 \sin 2 x+4 \cos 2 x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q7
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q7.1

Question 8.
\(\int \frac{1}{\cos x-\sin x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q8

Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B)

Question 9.
\(\int \frac{1}{\cos x-\sqrt{3} \sin x} \cdot d x\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Indefinite Integration Ex 3.2(B) II Q9

Class 12 Maharashtra State Board Maths Solution