Question

Find the antiderivative of the function $(\sin \frac{x}{2}+\cos \frac{x}{2}{)}^2.$

Answer 1

Anti - derivative of the function ${(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right))}^2$

=$\int {(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right))}^{2}dx$

$=\int ({\sin }^2\left(\frac{x}{2}\right)+{\cos }^2\left(\frac{x}{2}\right)+2\sin \frac{x}{2}\cos \frac{x}{2})dx$

$=\int (1+\sin x)dx$

$[\because {\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}=1$ and, $\sin x=2\sin \frac{x}{2}\cos \frac{x}{2}]$

$=x-\cos x+c$.