Question

Value of $\int \frac{dx}{\sqrt{x(a-x)}}$ is

• A. $2{\sin }^{-1}\sqrt{\frac{x}{a}}+c$
• B. $-2{\sin }^{-1}\sqrt{\frac{x}{a}}+c$
• C. $2{\sin }^{-1}\frac{\sqrt{x}}{a}+c$
• D. None of these

Answer 1

Answer: (A)

Let $x=a{\sin }^2\theta$
then $dx=2a\sin \theta \cos \theta d\theta$
$\therefore I=\int \frac{2a\sin \theta \cdot \cos \theta }{\sqrt{a{\sin }^2\theta \cdot a{\cos }^2\theta }}d\theta$
$=2\int d\theta =2\theta +c$
$=2{\sin }^{-1}(\sqrt{x/a})+c$