Question

Evalue: ${\int }_0^{\pi }(si{n}^2\left(\frac{x}{2}\right)-co{s}^2\left(\frac{x}{2}\right))dx$.

Answer 1

$I={\int }_0^{\pi }({\sin }^2\left(\frac{x}{2}\right)-{\cos }^2\left(\frac{x}{2}\right))dx$

${\int }_a^{b}f\left(x\right)={\int }_a^{b}f(a+b-x)dx$

$I={\int }_0^{\pi }({\sin }^2\left(\frac{\pi +x}{2}\right)-{\cos }^2\left(\frac{\pi +x}{2}\right))dx$

$I={\int }_0^{\pi }({\cos }^2\left(\frac{x}{2}\right)-{\sin }^2\left(\frac{x}{2}\right))dx$

$I+I={\int }_0^{\pi }({\sin }^2\left(\frac{x}{2}\right)-{\cos }^2\left(\frac{x}{2}\right)+{\cos }^2\left(\frac{x}{2}\right)-{\sin }^2\left(\frac{x}{2}\right))dx$

$=0$

$\therefore I=0$