Question

Integrate : $\int \frac{1-\sin 2x}{x+{\cos }^{2}x}dx$

Answer 1

Given : $\int \frac{1-\sin 2x}{x+{\cos }^{2}x}dx$

Let $I=\int \frac{1-\sin 2x}{x+{\cos }^{2}x}dx$

$t=x+{\cos }^{2}x$

$\Rightarrow dt=(1-2\cos x\sin x)dx=(1-\sin 2x)dx$

$\int \frac{(1-\sin 2x)dx}{x+{\cos }^{2}x}$$=\int \frac{dt}{t}$

$I=\log \left|t\right|+c$

Where $c$ is constant of integration and $t=x+{\cos }^{2}x$

$I=\log |x+{\cos }^{2}x|+c$