Question

Solve
$J={\int }_o^{7/2}\sqrt{1+sinx}dx.$

Answer 1

$I={\int }_0^{7/2}\sqrt{1+\sin x}dx$

$={\int }_0^{7/2}\sqrt{{\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}+2\sin \frac{x}{2}.\cos \frac{x}{2}}dx$

$={\int }_0^{7/2}(\sin \frac{x}{2}+\cos \frac{x}{2})dx$

$=-2\cos \frac{x}{2}+2\sin \frac{x}{2}{\int }_0^{7/2}$

$=-2\cos \frac{7}{4}+2\sin \frac{7}{4}+2=0$

$=+2[-\cos \frac{7}{4}+\sin \frac{7}{4}+1]$