MCQ
$\int_0^{1/\sqrt 2 } {\frac{{{{\sin }^{ - 1}}x}}{{{{(1 - {x^2})}^{3/2}}}}dx = } $
- A$\frac{\pi }{4} + \frac{1}{2}\log 2$
- ✓$\frac{\pi }{4} - \frac{1}{2}\log 2$
- C$\frac{\pi }{2} + \log 2$
- D$\frac{\pi }{2} - \log 2$
Put ${\sin ^{ - 1}}x = t$
$\Rightarrow \frac{1}{{\sqrt {1 - {x^2}} }}dx = dt$ and $x = \sin t$
Also $t = 0$ to $\frac{\pi }{4}$
as $x = 0$ to $\frac{1}{{\sqrt 2 }}$
$ \Rightarrow I = \int_0^{\pi /4} {t.{{\sec }^2}t\,dt = \frac{\pi }{4} - \frac{1}{2}\log 2} $.
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