Question:
Evaluate $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x$
Solution:
Let, $x=\sin ^{2} t$
Differentiating both side with respect to t
$\frac{d x}{d t}=2 \sin t \cos t \Rightarrow d x=2 \sin t \cos t d t$
$y=\int \sqrt{\frac{1-\sin t}{1+\sin t}} 2 \sin t \cos t d t$
$y=\int \sqrt{\frac{(1-\sin t)}{(1+\sin t)} \times \frac{(1-\sin t)}{(1-\sin t)}} 2 \sin t \cos t d t$
$y=2 \int(1-\sin t) \sin t d t$
$y=2 \int \sin t-\frac{1-\cos 2 t}{2} d t$
$y=2\left(-\cos t-\frac{t}{2}+\frac{\sin 2 t}{4}\right)+c$
Again, put $t=\sin \sqrt{x}$
$y=2\left(-\cos \sin \sqrt{x}-\frac{\sin \sqrt{x}}{2}+\frac{\sin (2 \sin \sqrt{x})}{4}\right)+c$
$y=2\left(-\sqrt{1-x}-\frac{\sin \sqrt{x}}{2}+\frac{1}{2} \sqrt{x-x^{2}}\right)+c$