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

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

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

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) $$\log \left(\frac{189}{208}\right)$$
(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) $$\frac{4+\pi}{2}$$
(a) $$\frac{4-\pi}{2}$$

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
(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) $$\frac{-5 \pi}{256}$$
(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
(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) $$\frac{\sqrt{e}-1}{e}$$
(c) √e(√e − 1)

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
(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
(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
(b) $$\frac{9}{2}$$

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
(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 = $$\frac{3}{10}$$

Question 2.
$$\int_{\pi / 4}^{\pi / 2} \frac{\cos \theta}{\left[\cos \frac{\theta}{2}+\sin \frac{\theta}{2}\right]^{3}} d \theta$$
Solution:

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

Question 4.
$$\int_{0}^{\pi / 4} \frac{\tan ^{3} x}{1+\cos 2 x} d x$$
Solution:

Question 5.
$$\int_{0}^{1} t^{5} \sqrt{1-t^{2}} d t$$
Solution:

Question 6.
$$\int_{0}^{1}\left(\cos ^{-1} x\right)^{2} d x$$
Solution:

Question 7.
$$\int_{-1}^{1} \frac{1+x^{3}}{9-x^{2}} d x$$
Solution:

Question 8.
$$\int_{0}^{\pi} x \cdot \sin x \cdot \cos ^{4} x d x$$
Solution:

Question 9.
$$\int_{0}^{\pi} \frac{x}{1+\sin ^{2} x} d x$$
Solution:

Question 10.
$$\int_{1}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x$$
Solution:

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:

Question 2.
$$\int_{0}^{\pi / 2} \frac{1}{6-\cos x} d x$$
Solution:

Question 3.
$$\int_{0}^{a} \frac{1}{a^{2}+a x-x^{2}} d x$$
Solution:

Question 4.
$$\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x}{\sin x+\cos x} d x$$
Solution:

Question 5.
$$\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$$
Solution:
Let I = $$\int_{0}^{1} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$$

Question 6.
$$\int_{0}^{\pi / 4} \frac{\cos 2 x}{1+\cos 2 x+\sin 2 x} d x$$
Solution:

Question 7.
$$\int_{0}^{\pi / 2}[2 \log (\sin x)-\log (\sin 2 x)] d x$$
Solution:

Question 8.
$$\int_{0}^{\pi}\left(\sin ^{-1} x+\cos ^{-1} x\right)^{3} \sin ^{3} x d x$$
Solution:

Question 9.
$$\int_{0}^{4}\left[\sqrt{x^{2}+2 x+3}\right]^{-1} d x$$
Solution:

Question 10.
$$\int_{-2}^{3}|x-2| d x$$
Solution:
|x – 2|= 2 – x, if x < 2
= x – 2, if x ≥ 2

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:

Question 2.
If $$\int_{0}^{k} \frac{1}{2+8 x^{2}} \cdot d x=\frac{\pi}{16}$$, find k.
Solution:

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: