Evaluate the following integrals:

$\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x$

Put $x=\operatorname{atan}^{2} t ; d x=2 a \cdot \tan t \cdot \sec ^{2} t d t$

$=\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x=\int \sin ^{-1} \sqrt{\frac{\mathrm{a} \tan ^{2} t}{a+\mathrm{a} \tan ^{2} t}} 2 \mathrm{a} \cdot \operatorname{tant} \cdot \sec ^{2} t \mathrm{dt}=\int \mathrm{t} \cdot 2 \mathrm{a} \cdot \tan \mathrm{t} \cdot \sec ^{2} t \mathrm{dt}$

$=2 a \int \mathrm{t} \cdot \tan \mathrm{t} \cdot \sec ^{2} t \mathrm{dt}$

$=2 a\left[\frac{t\left(\tan ^{2} t\right)}{2}-\int \frac{\tan ^{2} t}{2} d t\right]+c$

$=2 a\left[\frac{t\left(\tan ^{2} t\right)}{2}-\frac{\tan t}{2}+\frac{t}{2}\right]+c$

$=a\left[t\left(\tan ^{2} t\right)-\tan t+t\right]+c$

$=x \tan ^{-1} \sqrt{\frac{x}{a}}-\sqrt{a x}+\operatorname{atan}^{-1} \sqrt{\frac{x}{a}}+c$