Q111 / 160 · AP EAMCET 2024 (23 May Shift 1) · Mathematics
$\int \sin ^{-1}\left(\sqrt{\frac{x-a}{x}}\right) d x=$
- Correct answerA$x \cos ^{-1} \sqrt{\frac{a}{x}}-\sqrt{a x-a^2}+c$
- B$x \sec ^{-1} \sqrt{\frac{a}{x}}+\sqrt{x^2-a x}+c$
- C$x \sin ^{-1} \sqrt{\frac{x}{a}}+\sqrt{x^2+a x}+c$
- D$\frac{x}{a} \sin ^{-1} \frac{x}{a}+\frac{x^2}{a} \sqrt{1+a^2}+c$
Solution
$I=\int \sin ^{-1}\left(\sqrt{\frac{x-a}{x}}\right) d x$
$\begin{aligned} & \text { Let } x=a \sec ^2 \theta \Rightarrow d x=2 a \sec ^2 \theta \tan \theta \mathrm{~d} \theta \\ & I=\int \sin ^{-1}\left(\sqrt{\frac{a \sec ^2 \theta-a}{a \sec ^2 \theta}}\right) \cdot 2 a \sec ^2 \theta \tan \theta d \theta \\ & \Rightarrow I=\int \sin ^{-1}\left(\frac{\tan \theta}{\sec \theta}\right) \cdot 2 a \sec ^2 \theta \tan \theta d \theta \\ & \Rightarrow I=\int \theta \cdot 2 a \sec ^2 \theta \tan \theta d \theta=\theta a \sec ^2 \theta-a \int \sec ^2 \theta d \theta \\ & \Rightarrow I=\theta a \sec ^2 \theta-a \tan \theta+C=x \cos ^{-1} \sqrt{\frac{a}{x}}-\sqrt{a x-a^2}+C\end{aligned}$
Hence, the correct answer is option: 'A'.