Definite Integration Question 299
Question: $ \int_0^{1/\sqrt{2}}{\frac{{{\sin }^{-1}}x}{{{(1-x^{2})}^{3/2}}}dx=} $
[Roorkee 1984]
Options:
A) $ \frac{\pi }{4}+\frac{1}{2}\log 2 $
B) $ \frac{\pi }{4}-\frac{1}{2}\log 2 $
C) $ \frac{\pi }{2}+\log 2 $
D) $ \frac{\pi }{2}-\log 2 $
Show Answer
Answer:
Correct Answer: B
Solution:
$ I=\int_0^{1/\sqrt{2}}{\frac{{{\sin }^{-1}}x}{{{(1-x^{2})}^{3/2}}}}dx $
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}}tdt=\frac{\pi }{4}-\frac{1}{2}\log 2} $ .
BETA
