Question

Evaluate $\sqrt[\pi ]{\pi 0}[si{n}^2\frac{\pi }{2}-Co{s}^2\frac{x}{2}]dx$.

Answer 1

$\pi \sqrt{0}[{\sin }^2\frac{\pi }{2}-\cos \frac{x}{2}]dx$

$=\int {\sin }^2\frac{\pi }{2}dx-\int {\cos }^2\frac{x}{2}dx$

$\therefore \int \frac{1-\cos 2\pi /2}{2}dx-\int \frac{1+\cos 2\pi /2}{2}dx$

$=\int \frac{1}{2}-\frac{\cos \pi }{2}dx-\int \frac{1}{2}+\frac{\cos \pi }{2}dx$

$=\frac{x}{2}-\frac{\sin \pi }{2}-\frac{x}{2}+\frac{\sin \pi }{4}$

$=\frac{\sin x}{4}-\frac{\sin \pi }{4}+C$