Indefinite Integration 1 Question 84
85. Evaluate $\int _0^{1 / 2} \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$.
$(1984,2 M)$
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Answer:
Correct Answer: 85. $-\frac{\sqrt{3}}{12} \pi+\frac{1}{2}$
Solution:
- Let $I=\int _0^{1 / 2} \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$ Put $^{-1} x=\theta \Rightarrow x=\sin \theta$
$\Rightarrow d x=\cos \theta d \theta$
$\therefore I=\int _0^{\pi / 6} \frac{\theta \sin \theta}{\sqrt{1-\sin ^{2} \theta}} \cdot \cos \theta d \theta=\int _0^{\pi / 6} \theta \sin \theta d \theta$
$=[-\theta \cos \theta] _0^{\pi / 6}+\int _0^{\pi / 6} \cos \theta d \theta$
$=-\frac{\pi}{6} \cos \frac{\pi}{6}+0+\sin \frac{\pi}{6}-\sin 0=-\frac{\sqrt{3} \pi}{12}+\frac{1}{2}$
