Q69 / 100 · BITSAT 2023 (Memory Based Paper 2) · Mathematics
Value of $\int \frac{d x}{\sqrt{x(a-x)}}$ is
- Correct answerA$2 \sin ^{-1} \sqrt{\frac{x}{a}}+c$
- B$-2 \sin ^{-1} \sqrt{\frac{x}{a}}+c$
- C$2 \sin ^{-1} \frac{\sqrt{x}}{a}+c$
- D
None of these
Solution
Let $\mathrm{x}=\mathrm{a} \sin ^2 \theta$
then $d x=2 a \sin \theta \cos \theta d \theta$
$\begin{aligned}<br />\therefore \quad I & =\int \frac{2 a \sin \theta \cdot \cos \theta}{\sqrt{a \sin ^2 \theta \cdot a \cos ^2 \theta}} d \theta \\<br />& =2 \int d \theta=2 \theta+c \\<br />& =2 \sin ^{-1}(\sqrt{x / a})+c<br />\end{aligned}$
then $d x=2 a \sin \theta \cos \theta d \theta$
$\begin{aligned}<br />\therefore \quad I & =\int \frac{2 a \sin \theta \cdot \cos \theta}{\sqrt{a \sin ^2 \theta \cdot a \cos ^2 \theta}} d \theta \\<br />& =2 \int d \theta=2 \theta+c \\<br />& =2 \sin ^{-1}(\sqrt{x / a})+c<br />\end{aligned}$
Hence, the correct answer is option: 'A'.