Question

Solve
${\int }_{-\pi }^{\pi }\frac{\sin x\cos }{1+{\cos }^{2}x}dx$

Answer 1

Consider the given integral.

$I={\int }_{-\pi }^{\pi }\frac{\sin x\cos x}{1+{\cos }^{2}x}dx$ $............\left(1\right)$

Now, apply property

$I=2{\int }_0^{\pi }\frac{\sin x\cos x}{1+{\cos }^{2}x}dx$ $............\left(1\right)$

$I=-2{\int }_0^{\pi }\frac{\sin x\cos x}{1+{\cos }^{2}x}dx$ $............\left(2\right)$

On adding equation (1) and (2), we get

$2I=0$

$I=0$

Hence, this is the answer.