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Std 12 Maths 1 Exercise 8.4 Solutions Commerce Maths

Solve the following differential equations:

Question 1.
x dx + 2y dy = 0
Solution:
x dx + 2y dy = 0
Integrating, we get
∫x dx + 2 ∫y dy = c1
∴ \(\frac{x^{2}}{2}+2\left(\frac{y^{2}}{2}\right)=c_{1}\)
∴ x2 + 2y2 = c, where c = 2c1
This is the general solution.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4

Question 2.
y2 dx + (xy + x2) dy = 0
Solution:
y2 dx + (xy + x2) dy = 0
∴ (xy + x2) dy = -y2 dx
∴ \(\frac{d y}{d x}=\frac{-y^{2}}{x y+x^{2}}\) ………(1)
Put y = vx
∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)
Substituting these values in (1), we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q2
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q2.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q2.2
This is the general solution.

Question 3.
x2y dx – (x3 + y3) dy = 0
Solution:
x2y dx – (x3 + y3) dy = 0
∴ (x3 + y3) dy = x2y dx
∴ \(\frac{d y}{d x}=\frac{x^{2} y}{x^{3}+y^{3}}\) ……(1)
Put y = vx
∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q3
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q3.1
This is the general solution.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4

Question 4.
\(\frac{d y}{d x}+\frac{x-2 y}{2 x-y}=0\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q4
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q4.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q4.2
This is the general solution.

Question 5.
(x2 – y2) dx + 2xy dy = 0
Solution:
(x2 – y2) dx + 2xy dy = 0
∴ 2xy dy = -(x2 – y2) dx = (y2 – x2) dx
∴ \(\frac{d y}{d x}=\frac{y^{2}-x^{2}}{2 x y}\) ………(1)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q5
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q5.1

Question 6.
xy\(\frac{d y}{d x}\) = x2 + 2y2
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q6
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q6.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4

Question 7.
x2\(\frac{d y}{d x}\) = x2 + xy – y2
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q7
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.4 Q7.1

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Question 1.
Solve the following differential equations:
(i) \(\frac{d y}{d x}\) = x2y + y
Solution:
\(\frac{d y}{d x}\) = x2y + y
∴ \(\frac{d y}{d x}\) = y(x2 + 1)
∴ \(\frac{1}{y}\) dy = (x2 + 1) dx
Integrating, we get
∫\(\frac{1}{y}\) dy = ∫(x2 + 1) dx
∴ log |y|= \(\frac{x^{3}}{3}\) + x + c
This is the general solution.

(ii) \(\frac{d \theta}{d t}=-k\left(\theta-\theta_{0}\right)\)
Solution:
\(\frac{d \theta}{d t}=-k\left(\theta-\theta_{0}\right)\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q1(ii)
This is the general solution.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3

(iii) (x2 – yx2) dy + (y2 + xy2) dx = 0
Solution:
(x2 – yx2) dy + (y2 + xy2) dx = 0
∴ x2(1 – y) dy + y2(1 + x) dx = 0
∴ \(\frac{1-y}{y^{2}} d y+\frac{1+x}{x^{2}} d x=0\)
Integrating, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q1(iii)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q1(iii).1
This is the general solution.

(iv) \(y^{3}-\frac{d y}{d x}=x \frac{d y}{d x}\)
Solution:
\(y^{3}-\frac{d y}{d x}=x \frac{d y}{d x}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q1(iv)
∴ 2y2 log |x + 1| = 2cy2 – 1 is the required solution.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3

Question 2.
For each of the following differential equations find the particular solution:
(i) (x – y2x) dx – (y + x2y) dy = 0, when x = 2, y = 0.
Solution:
(x – y2x) dx – (y + x2y) dy = 0
∴ x(1 – y2) dx – y(1 + x2) dy = 0
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q2(i)
∴ the general solution is
log |1 + x2| + log |1 – y2| = log c, where c1 = log c
∴ log |(1 + x2)(1 – y2) | = log c
∴ (1 + x2)(1 – y2) = c
When x = 2, y = 0, we have
(1 + 4)(1 – 0) = c
∴ c = 5
∴ the particular solution is (1 + x2)(1 – y2) = 5.

(ii) (x + 1) \(\frac{d y}{d x}\) -1 = 2e-y, when y = 0, x = 1.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q2(ii)
∴ log |2 + ey| = log |c(x + 1)|
∴ 2 + ey = c(x + 1)
This is the general solution.
Now, y = 0, when x = 1
∴ 2 + e0 = c(1 + 1)
∴ 3 = 2c
∴ c = \(\frac{3}{2}\)
∴ the particular solution is
2 + ey = \(\frac{3}{2}\)(x + 1)
∴ 4 + 2ey = 3x + 3
∴ 3x – 2ey – 1 = 0

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3

(iii) y(1 + log x) \(\frac{d x}{d y}\) – x log x = 0, when x = e, y = e2.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q2(iii)
∴ from (1), the general solution is
log |x log x| – log |y| = log c, where c1 = log c
∴ log |\(\frac{x \log x}{y}\)| = log c
∴ \(\frac{x \log x}{y}\) = c
∴ x log x = cy
This is the general solution.
Now, y = e2, when x = e
e log e = ce2
1 = ce ……[∵ log e = 1]
c = \(\frac{1}{e}\)
∴ the particular solution is x log x = (\(\frac{1}{e}\)) y
∴ y = ex log x

(iv) \(\frac{d y}{d x}\) = 4x + y + 1, when y = 1, x = 0.
Solution:
\(\frac{d y}{d x}\) = 4x + y + 1
Put 4x + y + 1 = v
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.3 Q2(iv)
∴ log |v + 4| = x + c
∴ log |4x + y + 1 + 4| = x + c
i.e. log |4x + y + 5| = x + c
This is the general solution.
Now, y = 1 when x = 0
∴ log|0 + 1 + 5| = 0 + c,
i.e. c = log 6
∴ the particular solution is
log |4x + y + 5| = x + log 6
∴ \(\log \left|\frac{4 x+y+5}{6}\right|\) = x

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Question 1.
Obtain the differential equation by eliminating arbitrary constants from the following equations:
(i) y = Ae3x + Be-3x
Solution:
y = Ae3x + Be-3x ……(1)
Differentiating twice w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(i)
This is the required D.E.

(ii) y = \(c_{2}+\frac{c_{1}}{x}\)
Solution:
y = \(c_{2}+\frac{c_{1}}{x}\)
∴ xy = c2x + c1
Differentiating w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(ii)

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2

(iii) y = (c1 + c2x) ex
Solution:
y = (c1 + c2x) ex
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(iii)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(iii).1
This is the required D.E.

(iv) y = c1 e3x+ c2 e2x
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(iv)
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(iv).1
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(iv).2
This is the required D.E.

(v) y2 = (x + c)3
Solution:
y2 = (x + c)3
Differentiating w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q1(v)
This is the required D.E.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2

Question 2.
Find the differential equation by eliminating arbitrary constant from the relation x2 + y2 = 2ax.
Solution:
x2 + y2 = 2ax
Differentiating both sides w.r.t. x, we get
2x + 2y\(\frac{d y}{d x}\) = 2a
Substituting value of 2a in equation (1), we get
x2 + y2 = [2x + 2y \(\frac{d y}{d x}\)]x = 2x2 + 2xy \(\frac{d y}{d x}\)
∴ 2xy \(\frac{d y}{d x}\) = y2 – x2 is the required D.E.

Question 3.
Form the differential equation by eliminating arbitrary constants from the relation bx + ay = ab.
Solution:
bx + ay = ab
∴ ay = -bx + ab
∴ y = \(-\frac{b}{a} x+b\)
Differentiating w.r.t. x, we get
\(\frac{d y}{d x}=-\frac{b}{a} \times 1+0=-\frac{b}{a}\)
Differentiating again w.r.t. x, we get
\(\frac{d^{2} y}{d x^{2}}\) = 0 is the required D.E.

Question 4.
Find the differential equation whose general solution is x3 + y3 = 35ax.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2 Q4

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2

Question 5.
Form the differential equation from the relation x2 + 4y2 = 4b2.
Sol ution:
x2 + 4y2 = 4b2
Differentiating w.r.t. x, we get
2x + 4(2y\(\frac{d y}{d x}\)) = 0
i.e. x + 4y\(\frac{d y}{d x}\) = 0 is the required D.E.

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Question 1.
Determine the order and degree of each of the following differential equations:
(i) \(\frac{d^{2} x}{d t^{2}}+\left(\frac{d x}{d t}\right)^{2}+8=0\)
Solution:
The given D.E. is \(\frac{d^{2} x}{d t^{2}}+\left(\frac{d x}{d t}\right)^{2}+8=0\)
This D.E. has highest order derivative \(\frac{d^{2} x}{d t^{2}}\) with power 1.
∴ the given D.E. is of order 2 and degree 1.

(ii) \(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\left(\frac{d y}{d x}\right)^{2}=a^{x}\)
Solution:
The given D.E. is \(\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\left(\frac{d y}{d x}\right)^{2}=a^{x}\)
This D.E. has highest order derivative \(\frac{d^{2} y}{d x^{2}}\) with power 2.
∴ the given D.E. is of order 2 and degree 2.

(iii) \(\frac{d^{4} y}{d x^{4}}+\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3}\)
Solution:
The given D.E. is \(\frac{d^{4} y}{d x^{4}}+\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3}\)
This D.E. has highest order derivative \(\frac{d^{4} y}{d x^{4}}\) with power 1.
∴ the given D.E. is of order 4 and degree 1.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1

(iv) (y'”)2 + 2(y”)2 + 6y’ + 7y = 0
Solution:
The given D.E. is (y”‘)2 + 2(y”)2 + 6y’ + 7y = 0
This can be written as \(\left(\frac{d^{3} y}{d x^{3}}\right)^{2}+2\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+6 \frac{d y}{d x}+7 y=0\)
This D.E. has highest order derivative \(\frac{d^{3} y}{d x^{3}}\) with power 2.
∴ the given D.E. is of order 3 and degree 2.

(v) \(\sqrt{1+\frac{1}{\left(\frac{d y}{d x}\right)^{2}}}=\left(\frac{d y}{d x}\right)^{3 / 2}\)
Solution:
The given D.E. is \(\sqrt{1+\frac{1}{\left(\frac{d y}{d x}\right)^{2}}}=\left(\frac{d y}{d x}\right)^{3 / 2}\)
On squaring both sides, we get
\(1+\frac{1}{\left(\frac{d y}{d x}\right)^{2}}=\left(\frac{d y}{d x}\right)^{3}\)
∴ \(\left(\frac{d y}{d x}\right)^{2}+1=\left(\frac{d y}{d x}\right)^{5}\)
This D.E. has highest order derivative \(\frac{d y}{d x}\) with power 5.
∴ the given D.E. is of order 1 and degree 5.

(vi) \(\frac{d y}{d x}=7 \frac{d^{2} y}{d x^{2}}\)
Solution:
The given D.E. is \(\frac{d y}{d x}=7 \frac{d^{2} y}{d x^{2}}\)
This D.E. has highest order derivative \(\frac{d^{2} y}{d x^{2}}\) with power 1.
∴ the given D.E. is of order 2 and degree 1.

(vii) \(\left(\frac{d^{3} y}{d x^{3}}\right)^{1 / 6}=9\)
Solution:
The given D.E. is \(\left(\frac{d^{3} y}{d x^{3}}\right)^{1 / 6}=9\)
i.e., \(\frac{d^{3} y}{d x^{3}}=9^{6}\)
This D.E. has highest order derivative \(\frac{d^{3} y}{d x^{3}}\) with power 1.
∴ the given D.E. is of order 3 and degree 1.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1

Question 2.
In each of the following examples, verify that the given function is a solution of the corresponding differential equation:
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1 Q2
Solution:
(i) xy = log y + k
Differentiating w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1 Q2(i)
Hence, xy = log y + k is a solution of the D.E. y'(1 – xy) = y2.

(ii) y = xn
Differentiating twice w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1 Q2(ii)
This shows that y = xn is a solution of the D.E.
\(x^{2} \frac{d^{2} y}{d x^{2}}-n x \frac{d y}{d x}+n y=0\)

(iii) y = ex
Differentiating w.r.t. x, we get
\(\frac{d y}{d x}\) = ex = y
Hence, y = ex is a solution of the D.E. \(\frac{d y}{d x}\) = y.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1

(iv) y = 1 – log x
Differentiating w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1 Q2(iv)
Hence, y = 1 – log x is a solution of the D.E.
\(x^{2} \frac{d^{2} y}{d x^{2}}=1\)

(v) y = aex + be-x
Differentiating w.r.t. x, we get
\(\frac{d y}{d x}\) = a(ex) + b(-e-x) = aex – be-x
Differentiating again w.r.t. x, we get
\(\frac{d^{2} y}{d x^{2}}\) = a(ex) – b(-e-x)
= aex + be-x
= y
Hence, y = aex + be-x is a solution of the D.E. \(\frac{d^{2} y}{d x^{2}}\) = y.

Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1

(vi) ax2 + by2 = 5
Differentiating w.r.t. x, we get
Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.1 Q2(vi)
Hence, ax2 + by2 = 5 is a solution of the D.E.
\(x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}=y\left(\frac{d y}{d x}\right)\)

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(I) Choose the correct alternatives:

Question 1.
Area of the region bounded by the curve x2 = y, the X-axis and the lines x = 1 and x = 3 is
(a) \(\frac{26}{3}\) sq units
(b) \(\frac{3}{26}\) sq units
(c) 26 sq units
(d) 3 sq units
Answer:
(a) \(\frac{26}{3}\) sq units

Question 2.
The area of the region bounded by y2 = 4x, the X-axis and the lines x = 1 and x = 4 is
(a) 28 sq units
(b) 3 sq unit
(c) \(\frac{28}{3}\) sq units
(d) \(\frac{3}{28}\) sq units
Answer:
(c) \(\frac{28}{3}\) sq units

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 3.
Area of the region bounded by x2 = 16y, y = 1 and y = 4 and the Y-axis, lying in the first quadrant is
(a) 63 sq units
(b) \(\frac{3}{56}\) sq units
(c) \(\frac{56}{3}\) sq units
(d) \(\frac{63}{7}\) sq units
Answer:
(c) \(\frac{56}{3}\) sq units

Question 4.
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is
(a) \(\frac{3142}{5}\) sq units
(b) \(\frac{3124}{5}\) sq units
(c) \(\frac{3142}{3}\) sq units
(d) \(\frac{3124}{3}\) sq units
Answer:
(b) \(\frac{3124}{5}\) sq units

Question 5.
Using definite integration area of circle x2 + y2 = 25 is
(a) 5π sq units
(b) 4π sq units
(c) 25π sq units
(d) 25 sq units
Answer:
(c) 25π sq units

(II) Fill in the blanks:

Question 1.
Area of the region bounded by y = x4, x = 1, x = 5 and the X-axis is _________
Answer:
\(\frac{3124}{5}\) sq units

Question 2.
Using definite integration area of the circle x2 + y2 = 49 is ___________
Answer:
49π sq units

Question 3.
Area of the region bounded by x2 = 16y, y = 1, y = 4 and the Y-axis lying in the first quadrant is _________
Answer:
\(\frac{56}{3}\) sq units

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 4.
The area of the region bounded by the curve x2 = y, the X-axis and the lines x = 3 and x = 9 is _________
Answer:
234 sq units

Question 5.
The area of the region bounded by y2 = 4x, the X-axis and the lines x = 1 and x = 4 is __________
Answer:
\(\frac{28}{3}\) sq units

(III) State whether each of the following is True or False.

Question 1.
The area bounded by the curve x = g(y), Y-axis and bounded between the lines y = c and y = d is given by \(\int_{c}^{d} x d y=\int_{y=c}^{y=d} g(y) d y\)
Answer:
True

Question 2.
The area bounded by two curves y = f(x), y = g(x) and X-axis is \(\left|\int_{a}^{b} f(x) d x-\int_{b}^{a} g(x) d x\right|\)
Answer:
False

Question 3.
The area bounded by the curve y = f(x), X-axis and lines x = a and x = b is \(\left|\int_{a}^{b} f(x) d x\right|\)
Answer:
True

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 4.
If the curve, under consideration, is below the X-axis, then the area bounded by curve, X-axis, and lines x = a, x = b is positive.
Answer:
False

Question 5.
The area of the portion lying above the X-axis is positive.
Answer:
True

(IV) Solve the following:

Question 1.
Find the area of the region bounded by the curve xy = c2, the X-axis, and the lines x = c, x = 2c.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q1
= c2 log(\(\frac{2 c}{c}\))
= c2 . log 2 sq units.

Question 2.
Find the area between the parabolas y2 = 7x and x2 = 7y.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q2
For finding the points of intersection of the two parabolas,
we equate the values of y2 from their equations.
From the equation x2 = 7y, y2 = \(\frac{x^{4}}{49}\)
∴ \(\frac{x^{4}}{49}\) = 7x
∴ x4 = 343x
∴ x4 – 343x = 0
∴ x(x3 – 343) = 0
∴ x = 0 or x3 = 343, i.e. x = 7
When x = 0, y = 0
When x = 7, 7y = 49
∴ y = 7
∴ the points of intersection are O(0, 0) and A(7, 7)
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 = 7x
i.e. y = √7 √x
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q2.1
Area of the region ODABO = Area under the parabola
x2 = 7y
i.e. y = \(\frac{x^{2}}{7}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q2.2
∴ required area = \(\frac{98}{3}-\frac{49}{3}=\frac{49}{3}\) sq units.

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 3.
Find the area of the region bounded by the curve y = x2 and the line y = 10.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q3
By the symmetry of the parabola,
the required area is twice the area of the region OABCO
Now, the area of the region OABCO
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q3.1

Question 4.
Find the area of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}\) = 1.
Solution:
By the symmetry of the ellipse, the required area of the ellipse is 4 times the area of the region OPQO.
For the region OPQO, the limits of integration are x = 0 and x = 4.
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q4
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q4.1

Question 5.
Find the area of the region bounded by y = x2, the X-axis and x = 1, x = 4.
Solution:
Required area = \(\int_{1}^{4} y d x\), where y = x2
= \(\int_{1}^{4} x^{2} d x\)
= \(\left[\frac{x^{3}}{3}\right]_{1}^{4}=\frac{4^{3}}{3}-\frac{1}{3}=\frac{64-1}{3}\)
= 21 sq units.

Question 6.
Find the area of the region bounded by the curve x2 = 25y, y = 1, y = 4, and the Y-axis.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q6

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7

Question 7.
Find the area of the region bounded by the parabola y2 = 25x and the line x = 5.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q7
Given the equation of the parabola is y2 = 25x
∴ y = 5√x …… [∵ IIn first quadrant, y > 0]
Required area = area of the region OQRPO
= 2(area of the region ORPO)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Miscellaneous Exercise 7 IV Q7.1

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Question 1.
Find the area of the region bounded by the following curves, the X-axis, and the given lines:
(i) y = x4, x = 1, x = 5
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(i)

(ii) y = \(\sqrt{6 x+4}\), x = 0, x = 2
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(ii)

(iii) \(\sqrt{16-x^{2}}\), x = 0, x = 4
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(iii)

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

(iv) 2y = 5x + 7, x = 2, x = 8
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(iv)

(v) 2y + x = 8, x = 2, x = 4
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(v)

(vi) y = x2 + 1, x = 0, x = 3
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(vi)

(vii) y = 2 – x2, x = -1, x = 1
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q1(vii)

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

Question 2.
Find the area of the region bounded by the parabola y2 = 4x and the line x = 3.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q2
Required area = area of the region OABO
= 2(area of the region OACO)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q2.1

Question 3.
Find the area of the circle x2 + y2 = 25.
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q3
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 5.
From the equation of the circle, y2 = 25 – x2.
In the first quadrant y > 0
∴ y = \(\sqrt{25-x^{2}}\)
∴ area of the circle = 4(area of region OABO)
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q3.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1

Question 4.
Find the area of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{25}\) = 1
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q4
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 2.
From the equation of the ellipse,
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q4.1
Maharashtra Board 12th Commerce Maths Solutions Chapter 7 Application of Definite Integration Ex 7.1 Q4.2

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Std 12 Maths 1 Miscellaneous Exercise 6 Solutions Commerce Maths

(I) Choose the correct alternative:

Question 1.
\(\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x\) = ________
(a) 0
(b) 3
(c) 9
(d) -9
Answer:
(a) 0

Question 2.
\(\int_{-2}^{3} \frac{d x}{x+5}\) = _________
(a) -log(\(\frac{8}{3}\))
(b) log(\(\frac{8}{3}\))
(c) log(\(\frac{3}{8}\))
(d) -log(\(\frac{3}{8}\))
Answer:
(b) log(\(\frac{8}{3}\))

Question 3.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\) = _________
(a) log(\(\frac{8}{3}\))
(b) -log(\(\frac{8}{3}\))
(c) \(\frac{1}{2}\) log(\(\frac{8}{3}\))
(d) \(\frac{-1}{2}\) log(\(\frac{8}{3}\))
Answer:
(c) \(\frac{1}{2}\) log(\(\frac{8}{3}\))

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 4.
\(\int_{4}^{9} \frac{d x}{\sqrt{x}}\) = ___________
(a) 9
(b) 4
(c) 2
(d) 0
Answer:
(c) 2

Question 5.
If \(\int_{0}^{a} 3 x^{2} d x=8\), then a = __________
(a) 2
(b) 0
(c) \(\frac{8}{3}\)
(d) a
Answer:
(a) 2

Question 6.
\(\int_{2}^{3} x^{4}\) dx = ________
(a) \(\frac{1}{2}\)
(b) \(\frac{5}{2}\)
(c) \(\frac{5}{211}\)
(d) \(\frac{211}{5}\)
Answer:
(d) \(\frac{211}{5}\)

Question 7.
\(\int_{0}^{2} e^{x}\) dx = _______
(a) e – 1
(b) 1 – e
(c) 1 – e2
(d) e2 – 1
Answer:
(d) e2 – 1

Question 8.
\(\int_{a}^{b} f(x) d x\) = ________
(a) \(\int_{b}^{a} f(x) d x\)
(b) –\(\int_{a}^{b} f(x) d x\)
(c) –\(\int_{b}^{a} f(x) d x\)
(d) \(\int_{0}^{a} f(x) d x\)
Answer:
(c) –\(\int_{b}^{a} f(x) d x\)

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 9.
\(\int_{-7}^{7} \frac{x^{3}}{x^{2}+7} d x\) = _________
(a) 7
(b) 49
(c) 0
(d) \(\frac{7}{2}\)
Answer:
(c) 0

Question 10.
\(\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x\) = _________
(a) \(\frac{7}{2}\)
(b) \(\frac{5}{2}\)
(c) 7
(d) 2
Answer:
(b) \(\frac{5}{2}\)

(II) Fill in the blanks:

Question 1.
\(\int_{0}^{2} e^{x} d x\) = ________
Answer:
e2 – 1

Question 2.
\(\int_{2}^{3} x^{4} d x\) = __________
Answer:
\(\frac{211}{5}\)

Question 3.
\(\int_{0}^{1} \frac{d x}{2 x+5}\) = ____________
Answer:
\(\frac{1}{2} \log \left(\frac{7}{5}\right)\)

Question 4.
If \(\int_{0}^{a} 3 x^{2} d x\) = 8, then a = _________
Answer:
2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 5.
\(\int_{4}^{9} \frac{1}{\sqrt{x}} d x\) = _________
Answer:
2

Question 6.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\) = _________
Answer:
\(\frac{1}{2} \log \left(\frac{8}{3}\right)\)

Question 7.
\(\int_{-2}^{3} \frac{d x}{x+5}\) = _________
Answer:
\(\log \left(\frac{8}{3}\right)\)

Question 8.
\(\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x\) = _____________
Answer:
o

(III) State whether each of the following is True or False:

Question 1.
\(\int_{a}^{b} f(x) d x=\int_{-b}^{-a} f(x) d x\)
Answer:
True

Question 2.
\(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(t) d t\)
Answer:
True

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 3.
\(\int_{0}^{a} f(x) d x=\int_{a}^{0} f(a-x) d x\)
Answer:
False

Question 4.
\(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(x-a-b) d x\)
Answer:
False

Question 5.
\(\int_{-5}^{5} \frac{x^{3}}{x^{2}+7} d x=0\)
Answer:
True

Question 6.
\(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x=\frac{1}{2}\)
Answer:
True

Question 7.
\(\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x=\frac{9}{2}\)
Answer:
False

Question 8.
\(\int_{4}^{7} \frac{(11-x)^{2}}{(11-x)^{2}+x^{2}} d x=\frac{3}{2}\)
Answer:
True

(IV) Solve the following:

Question 1.
\(\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q1
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q1.1

Question 2.
\(\int_{1}^{2} \frac{x+3}{x(x+2)} d x\)
Solution:
Let I = \(\int_{1}^{2} \frac{x+3}{x(x+2)} d x\)
Let \(\frac{x+3}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2}\)
∴ x + 3 = A(x + 2) + Bx
Put x = 0, we get
3 = A(2) + B(0)
∴ A = \(\frac{3}{2}\)
Put x + 2 = 0, i.e. x = -2, we get
-2 + 3 = A(0) + B(-2)
∴ 1 = -2B
∴ B = \(-\frac{1}{2}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 3.
\(\int_{1}^{3} x^{2} \log x d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q3

Question 4.
\(\int_{0}^{1} e^{x^{2}} \cdot x^{3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q4
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q4.1

Question 5.
\(\int_{1}^{2} e^{2 x}\left(\frac{1}{x}-\frac{1}{2 x^{2}}\right) d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q5

Question 6.
\(\int_{4}^{9} \frac{1}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q6

Question 7.
\(\int_{-2}^{3} \frac{1}{x+5} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q7

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 8.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q8

Question 9.
\(\int_{0}^{1} \frac{x^{2}+3 x+2}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q9

Question 10.
\(\int_{3}^{5} \frac{d x}{\sqrt{x+4}+\sqrt{x-2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q10

Question 11.
\(\int_{2}^{3} \frac{x}{x^{2}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q11
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q11.1

Question 12.
\(\int_{1}^{2} x^{2} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q12

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 13.
\(\int_{-4}^{-1} \frac{1}{x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q13

Question 14.
\(\int_{0}^{1} \frac{1}{\sqrt{1+x}+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q14
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q14.1

Question 15.
\(\int_{0}^{4} \frac{1}{\sqrt{x^{2}+2 x+3}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q15
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q15.1

Question 16.
\(\int_{2}^{4} \frac{x}{x^{2}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q16

Question 17.
\(\int_{0}^{1} \frac{1}{2 x-3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q17

Question 18.
\(\int_{1}^{2} \frac{5 x^{2}}{x^{2}+4 x+3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q18
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q18.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6

Question 19.
\(\int_{1}^{2} \frac{d x}{x(1+\log x)^{2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q19

Question 20.
\(\int_{0}^{9} \frac{1}{1+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q20
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Miscellaneous Exercise 6 IV Q20.1

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Evaluate the following integrals:

Question 1.
\(\int_{-9}^{9} \frac{x^{3}}{4-x^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q1

Question 2.
\(\int_{0}^{a} x^{2}(a-x)^{3 / 2} d x\)
Solution:
We use the property
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2

Question 3.
\(\int_{1}^{3} \frac{\sqrt[3]{x+5}}{\sqrt[3]{x+5}+\sqrt[3]{9-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q3

Question 4.
\(\int_{2}^{5} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{7-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q4

Question 5.
\(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q5

Question 6.
\(\int_{2}^{7} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{9-x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q6

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2

Question 7.
\(\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q7
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q7.1

Question 8.
\(\int_{0}^{1} x(1-x)^{5} d x\)
Solution:
We use the property
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.2 Q8

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Evaluate the following definite integrals:

Question 1.
\(\int_{4}^{9} \frac{1}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q1

Question 2.
\(\int_{-2}^{3} \frac{1}{x+5} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q2

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 3.
\(\int_{2}^{3} \frac{x}{x^{2}-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q3
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q3.1

Question 4.
\(\int_{0}^{1} \frac{x^{2}+3 x+2}{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q4

Question 5.
\(\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x\)
Solution:
Let I = \(\int_{2}^{3} \frac{x}{(x+2)(x+3)} d x\)
Let \(\frac{x}{(x+2)(x+3)}=\frac{A}{x+3}+\frac{B}{x+2}\)
∴ x = A(x + 2) + B(x + 3)
Put x + 3 = 0, i.e. x = -3, we get
-3 = A(-1) + B(0)
∴ A = 3
Put x + 2 = 0, i.e. x = -2, we get
-2 = A(0) + B(1)
∴ B = -2
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q5
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q5.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 6.
\(\int_{1}^{2} \frac{d x}{x^{2}+6 x+5}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q6

Question 7.
If \(\int_{0}^{a}(2 x+1) d x\) = 2, find the real values of ‘a’.
Solution:
Let I = \(\int_{0}^{a}(2 x+1) d x\)
= \(\left[2 \cdot \frac{x^{2}}{2}+x\right]_{0}^{a}\)
= a2 + a – 0
= a2 + a
∴ I = 2 gives a2 + a = 2
∴ a2 + a – 2 = 0
∴ (a + 2)(a – 1) = 0 1
∴ a + 2 = 0 or a – 1 = 0
∴ a = -2 or a = 1.

Question 8.
If \(\int_{1}^{a}\left(3 x^{2}+2 x+1\right) d x\) = 11, find ‘a’.
Solution:
Let I = \(\int_{1}^{a}\left(3 x^{2}+2 x+1\right) d x\)
= \(\left[3\left(\frac{x^{3}}{3}\right)+2\left(\frac{x^{2}}{2}\right)+x\right]_{1}^{a}\)
= \(\left[x^{3}+x^{2}+x\right]_{1}^{a}\)
= (a3 + a2 + a) – (1 + 1 + 1)
= a3 + a2 + a – 3
∴ I = 11 gives a3 + a2 + a – 3 = 11
∴ a3 + a2 + a – 14 = 0
∴ (a3 – 8) + (a2 + a – 6) = 0
∴ (a – 2)(a2 + 2a + 4) + (a + 3)(a – 2) = 0
∴ (a – 2)(a2 + 2a + 4 + a + 3) = 0
∴ (a – 2)(a2 + 3a + 7) = 0
∴ a – 2 = 0 or a2 + 3a + 7 = 0
∴ a = 2 or a = \(\frac{-3 \pm \sqrt{9-28}}{2}\)
The latter two roots are not real.
∴ they are rejected.
∴ a = 2.

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 9.
\(\int_{0}^{1} \frac{1}{\sqrt{1+x}+\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q9

Question 10.
\(\int_{1}^{2} \frac{3 x}{9 x^{2}-1} d x\)
Solution:
Let I = \(\int_{1}^{2} \frac{3 x}{9 x^{2}-1} d x\) = \(\int_{1}^{2} \frac{3 x}{(3 x)^{2}-1} d x\)
Put 3x = t
∴ 3 dx = dt
∴ dx = \(\frac{d t}{3}\)
When x = 1, t = 3 × 1 = 3
When x = 2, t = 3 × 2 = 6
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q10
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q10.1

Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1

Question 11.
\(\int_{1}^{3} \log x d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 6 Definite Integration Ex 6.1 Q11

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Std 12 Maths 1 Miscellaneous Exercise 5 Solutions Commerce Maths

(I) Choose the correct alternative from the following:

Question 1.
The value of \(\int \frac{d x}{\sqrt{1-x}}\) is
(a) 20\(\sqrt{1-x}\) + c
(b) -2\(\sqrt{1-x}\) + c
(c) √x + c
(d) x + c
Answer:
(b) -2\(\sqrt{1-x}\) + c

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

Question 3.
\(\int x^{2}(3)^{x^{3}} d x\) =
(a) \(\text { (3) }^{x^{3}}+c\)
(b) \(\frac{(3)^{x^{3}}}{3 \cdot \log 3}+c\)
(c) \(\log 3(3)^{x^{3}}+c\)
(d) \(x^{2}(3)^{x 3}\)
Answer:
(b) \(\frac{(3)^{x^{3}}}{3 \cdot \log 3}+c\)
Hint:
Put x3 = t

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 4.
\(\int \frac{x+2}{2 x^{2}+6 x+5} d x=p \int \frac{4 x+6}{2 x^{2}+6 x+5} d x\) + \(\frac{1}{2} \int \frac{d x}{2 x^{2}+6 x+5}\), then p = __________
(a) \(\frac{1}{3}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{4}\)
(d) 2
Answer:
(c) \(\frac{1}{4}\)
Hint:
\(\int \frac{x+2}{2 x^{2}+6 x+5} d x=\int \frac{\frac{1}{4}(4 x+6)+\frac{1}{2}}{2 x^{2}+6 x+5} d x\)

Question 5.
\(\int \frac{d x}{\left(x-x^{2}\right)}\) = ________
(a) log x – log(1 – x) + c
(b) log(1 – x2) + c
(c) -log x + log(1 – x) + c
(d) log(x – x2) + c
Answer:
(a) log x – log(1 – x) + c

Question 6.
\(\int \frac{d x}{(x-8)(x+7)}\) = __________
(a) \(\frac{1}{15} \log \left|\frac{x+2}{x-1}\right|+c\)
(b) \(\frac{1}{15} \log \left|\frac{x+8}{x+7}\right|+c\)
(c) \(\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c\)
(d) (x – 8)(x – 7) + c
Answer:
(c) \(\frac{1}{15} \log \left|\frac{x-8}{x+7}\right|+c\)

Question 7.
\(\int\left(x+\frac{1}{x}\right)^{3} d x\) = _________
(a) \(\frac{1}{4}\left(x+\frac{1}{x}\right)^{4}+c\)
(b) \(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x-\frac{1}{2 x^{2}}+c\)
(c) \(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x+\frac{1}{x^{2}}+c\)
(d) \(\left(x-x^{-1}\right)^{3}+c\)
Answer:
(b) \(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+3 \log x-\frac{1}{2 x^{2}}+c\)
Hint:
\(\left(x+\frac{1}{x}\right)^{3}=x^{3}+3 x+\frac{3}{x}+\frac{1}{x^{3}}\)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 8.
\(\int\left(\frac{e^{2 x}+e^{-2 x}}{e^{x}}\right) d x\)
(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) \(e^{x}-\frac{1}{3 e^{3 x}}+c\)

Question 9.
∫(1 – x)-2 dx = ___________
(a) (1 + x)-1 + c
(b) (1 – x)-1 + c
(c) (1 – x)-1 – 1 + c
(d) (1 – x)-1 + 1 + c
Answer:
(b) (1 – x)-1 + c

Question 10.
\(\int \frac{\left(x^{3}+3 x^{2}+3 x+1\right)}{(x+1)^{5}} d x\) = _______
(a) \(\frac{-1}{x+1}+c\)
(b) \(\left(\frac{-1}{x+1}\right)^{5}+c\)
(c) log(x + 1) + c
(d) log|x + 1|5 + c
Answer:
(a) \(\frac{-1}{x+1}+c\)
Hint:
x3 + 3x2 + 3x + 1 = (x + 1)3

(II) Fill in the blanks.

Question 1.
\(\int \frac{5\left(x^{6}+1\right)}{x^{2}+1}\)dx = x4 + ___x3 + 5x + c.
Answer:
\(-\frac{5}{3}\)
Hint:
x6 + 1 = (x2 + 1)(x4 – x2 + 1)

Question 2.
\(\int \frac{x^{2}+x-6}{(x-2)(x-1)} d x\) = x + ______ + c
Answer:
4 log|x – 1|
Hint:
x2 + x – 6 = (x + 3)(x – 2)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 3.
If f'(x) = \(\frac{1}{x}\) + x and f(1) = \(\frac{5}{2}\) then f(x) = log x + \(\frac{x^{2}}{2}\) + _______
Answer:
2
Hint:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 II Q3

Question 4.
To find the value of \(\int \frac{(1+\log x) d x}{x}\) the proper substitution is __________
Answer:
1 + log x = t

Question 5.
\(\int \frac{1}{x^{3}}\left[\log x^{x}\right]^{2} d x\) = p(log x)3 + c, then p = _______
Answer:
\(\frac{1}{3}\)
Hint:
\(\frac{1}{x^{3}}\left(\log x^{x}\right)^{2}=\frac{1}{x^{3}}(x \log x)^{2}=\frac{(\log x)^{2}}{x}\)

(III) State whether each of the following is True or False:

Question 1.
The proper substitution for \(\int x\left(x^{x}\right)^{x}(2 \log x+1) d x \text { is }\left(x^{x}\right)^{x}=t\)
Answer:
True

Question 2.
If ∫x e2x dx is equal to e2x f(x) + c where c is constant of integration, then f(x) is \(\frac{(2 x-1)}{2}\).
Answer:
False

Question 3.
If ∫x f(x) dx = \(\frac{f(x)}{2}\), then f(x) = \(e^{x^{2}}\).
Answer:
True

Question 4.
If \(\int \frac{(x-1) d x}{(x+1)(x-2)}\) = A log|x + 1| + B log|x – 2|, then A + B = 1.
Answer:
True

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

Question 5.
For \(\int \frac{x-1}{(x+1)^{3}} e^{x} d x\) = ex f(x) + c, f(x) = (x + 1)2.
Answer:
False

(IV) Solve the following:

1. Evaluate:

(i) \(\int \frac{5 x^{2}-6 x+3}{2 x-3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(i)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(i).1

(ii) \(\int(5 x+1)^{\frac{4}{9}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(ii)

(iii) \(\int \frac{1}{2 x+3} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(iii)

(iv) \(\int \frac{x-1}{\sqrt{x+4}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(iv)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(iv).1

(v) If f'(x) = √x and f(1) = 2, then find the value of f(x).
Solution:
By the definition of integral
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q1(v)

(vi) ∫|x| dx if x < 0
Solution:
∫|x| dx = ∫-x dx …..[∵ x < 0]
= -∫x dx
= \(-\frac{x^{2}}{2}\) + c

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

2. Evaluate:

(i) Find the primitive of \(\frac{1}{1+e^{x}}\)
Solution:
Let I be the primitive of \(\frac{1}{1+e^{x}}\)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(i)

(ii) \(\int \frac{a e^{x}+b e^{-x}}{\left(a e^{x}-b e^{-x}\right)^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(ii)

(iii) \(\int \frac{1}{2 x+3 x \log x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(iii)

(iv) \(\int \frac{1}{\sqrt{x}+x} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(iv)

(v) \(\int \frac{2 e^{x}-3}{4 e^{x}+1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q2(v)

3. Evaluate:

(i) \(\int \frac{d x}{\sqrt{4 x^{2}-5}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(i)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(ii) \(\int \frac{d x}{3-2 x-x^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(ii)

(iii) \(\int \frac{d x}{9 x^{2}-25}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(iii)

(iv) \(\int \frac{e^{x}}{\sqrt{e^{2 x}+4 e^{x}+13}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(iv)

(v) \(\int \frac{d x}{x\left[(\log x)^{2}+4 \log x-1\right]}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(v)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(v).1

(vi) \(\int \frac{d x}{5-16 x^{2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(vi)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(vii) \(\int \frac{d x}{25 x-x(\log x)^{2}}\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(vii)

(viii) \(\int \frac{e^{x}}{4 e^{2 x}-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q3(viii)

4. Evaluate:

(i) ∫(log x)2 dx
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(i)

(ii) \(\int e^{x} \frac{1+x}{(2+x)^{2}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(ii)

(iii) ∫x e2x dx
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(iii)

(iv) ∫log(x2 + x) dx
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(iv)

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(v) \(\int e^{\sqrt{x}} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(v)

(vi) \(\int \sqrt{x^{2}+2 x+5} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(vi)

(vii) \(\int \sqrt{x^{2}-8 x+7} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q4(vii)

5. Evaluate:

(i) \(\int \frac{3 x-1}{2 x^{2}-x-1} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(i)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(i).1

Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5

(ii) \(\int \frac{2 x^{3}-3 x^{2}-9 x+1}{2 x^{2}-x-10} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(ii)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(ii).1
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(ii).2

(iii) \(\int \frac{1+\log x}{x(3+\log x)(2+3 \log x)} d x\)
Solution:
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(iii)
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(iii).1
Maharashtra Board 12th Commerce Maths Solutions Chapter 5 Integration Miscellaneous Exercise 5 IV Q5(iii).23

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