Evaluate: _0^ 1 / 2 ^ -1 x (1-x^2 )^ 3 / 2 d x - Vidyadip

Question

Evaluate: $\int_0^{1 / \sqrt{2}} \frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}} d x$

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Answer

Let $\sin ^{-1} x=\theta$ or, $x=\sin \theta$.
Then, $d x=d(\sin \theta)=\cos \theta d \theta$
Now, $ x=0$
$\Rightarrow \sin \theta=0$
$\Rightarrow \theta=0$ and $x=\frac{1}{\sqrt{2}}$
$\Rightarrow \sin \theta=\frac{1}{\sqrt{2}}$
$\Rightarrow \theta=\frac{\pi}{4}$
$\therefore I=\int_0^{1 / \sqrt{2}} \frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}} d x$
$\Rightarrow I=\int_0^{\pi / 4} \frac{\theta}{\cos ^3 \theta} \cos \theta d \theta$
$=\int_0^{\pi / 4} \theta \sec ^2 \theta d \theta$
Now using integratio by parts.
$\Rightarrow I=\left[\theta \tan \theta \right]_0^{\pi / 4}+[\log \cos \theta]_0^{\pi / 4}$
$=\frac{\pi}{4}+\left\{\log \left(\frac{1}{\sqrt{2}}\right)-\log 1\right\}$
$=\frac{\pi}{4}-\frac{1}{2} \log 2$

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