Question

Solve $\int \sqrt{1+\sin \frac{1}{2}x}dx$

Answer 1

$\int \sqrt{1+\sin \frac{1}{2}x}dx$

$=\int \sqrt{1+2\sin \frac{x}{4}\cos \frac{x}{4}}dx$

$=\int \sqrt{{\cos }^2\frac{x}{4}+{\sin }^2\frac{x}{4}+2\sin \frac{x}{4}\cos \frac{x}{4}}dx$

$=\int \sqrt{{(\cos \frac{1}{4}x+\sin \frac{1}{4}x)}^2}dx$

$=\int (\cos \frac{1}{4}x+\sin \frac{1}{4}x)dx$

$=4[\sin \frac{1}{4}x-\cos \frac{1}{4}x]+C$