Question

Evaluate the definite integral ${\int }_0^{\pi }({\sin }^2\frac{x}{2}-{\cos }^2\frac{x}{2})dx$

Answer 1

Let $I={\int }_0^{\pi }({\sin }^2\frac{x}{2}-{\cos }^2\frac{x}{2})dx$
$=-{\int }_0^{\pi }({\cos }^2\frac{x}{2}-si{n}^2\frac{x}{2})dx$
$=-{\int }_0^{\pi }\cos xdx$
$\Rightarrow \int \cos xdx=\sin x=F\left(x\right)$
By second fundamental theorem of calculus, we obtain
$I=F\left(\pi \right)-F\left(0\right)$
$=\sin \pi -\sin 0=0$