Question

Evaluate the following:

${\displaystyle \int \sqrt{1+\sin x}\text{d}x}$

Answer 1

I = ${\displaystyle \int \sqrt{1+\sin x}\text{d}x}$

= ${\displaystyle \int \sqrt{{\sin }^2 \frac{x}{2}+{\cos }^2 \frac{x}{2}+2\sin \frac{x}{2}\cos \frac{x}{2}}\text{d}x}$

= ${\displaystyle \int \sqrt{{(\sin \frac{x}{2}+\cos \frac{x}{2})}^2}\text{d}x}$

= ${\displaystyle \int (\sin \frac{x}{2}+\cos \frac{x}{2})\text{d}x}$

= ${\displaystyle \frac{1}{\frac{1}{2}}(-\cos \frac{x}{2}+\sin \frac{x}{2})+\text{C}}$

= ${\displaystyle 2(-\cos \frac{x}{2}+\sin \frac{x}{2})+\text{C}}$