Question

Evaluate: $\int\limits_{0}^{1 \sqrt{2}}\frac{sin^{-1}\,x}{\left(1-x^{2}\right)3/ 2} dx$

• A. $\frac{\pi}{4}- \frac{1}{2} log\,2$
• B. $\frac{\pi}{4}- log\,2$
• C. $\frac{\pi}{2}- log\,2$
• D. $\frac{\pi}{2}- \frac{1}{2} log\,2$

Answer 1

Answer: (A)

Let $sin^{-1} x = \theta \Rightarrow x = sin\,\theta. Then,$

$ dx = cos\,\theta \,d\theta $

Now, $x = 0 \Rightarrow \theta= 0$

and $x=\frac{1}{\sqrt{2}} \Rightarrow \theta =\frac{\pi}{4}$

$\therefore I=\int\limits_{0}^{\frac{1}{\sqrt{2}}} \frac{sin^{-1}x}{\left(1-x^{2}\right)^{\frac{3}{2}}} dx = \int\limits_{0}^{\frac{\pi}{4}}\frac{\theta cos\,\theta \,d\theta }{\left(cos ^{2}\theta\right) ^{\frac{3}{2}}}$

$ = \int\limits_{0}^{\frac{\pi }{4}}\theta \,sec^{2}\theta\, d\theta =\left[\theta\, tan \,\theta\right]_{0}^{\frac{\pi}{4}} -\int_{0}^{\frac{\pi}{4}} 1\cdot tan\, \theta \,d\theta $

$ = \left(\frac{\pi}{4}-0\right)+\left[log\, cos\,\theta\right]_{0}^{\frac{\pi}{4}} =\frac{\pi}{4} -\frac{1}{2} \,log\,2$