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

MCQ

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

A
$\frac{\pi}{4}+\frac{1}{2} \log 2$
✓
$\frac{\pi}{4}-\frac{1}{2} \log 2$
C
$\frac{\pi}{2}+\log 2$
D
$\frac{\pi}{2}-\log 2$

✓

Answer

Correct option: B.

$\frac{\pi}{4}-\frac{1}{2} \log 2$

(B)
Let $I =\int_0^{1 / \sqrt{2}} \frac{\sin ^{-1} x}{\left(1-x^2\right)^{\frac{3}{2}}} d x$
Put $\sin ^{-1} x= t \Rightarrow \frac{1}{\sqrt{1-x^2}} d x= dt$
When $x=0, t =0$ and when $x=\frac{1}{\sqrt{2}}, t =\frac{\pi}{4}$
$\therefore \quad I=\int_0^{\pi / 4} t \cdot \sec ^2 t d t=\frac{\pi}{4}-\frac{1}{2} \log 2$

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