Question

Solve $\int \frac{\sqrt{1-\sqrt{x}}}{\sqrt{1+\sqrt{x}}}.dx$

Answer 1

$I=\int \frac{\sqrt{1-\sqrt{x}}}{\sqrt{1+\sqrt{x}}}dx$

Multiply $\sqrt{1-\sqrt{x}}$

$I=\int \frac{1-\sqrt{x}}{\sqrt{1-x}}dx$

Putting $\sqrt{x}=\sin \theta$

$x={\sin }^2\theta$

$dx=2\sin \theta \cos \theta d\theta$

$I=\int \frac{1-\sin \theta }{\sqrt{1-{\sin }^2\theta }}2\sin \theta \cos \theta d\theta$

$I=\int \frac{1-\sin \theta }{\cos \theta }2\sin \theta \cos \theta d\theta$

$I=2\int (\sin \theta -{\sin }^2\theta )d\theta$

$I=2\int (\sin \theta -\frac{(\cos 2\theta -1)}{2})d\theta$

$I=2(-\cos \theta )-\frac{1}{2}\sin 2\theta +\theta +c$