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