Question

Solve : $\int \frac{\sin x\cos x}{{\cos }^{2}x+3\cos x+2}$

Answer 1

$\int \frac{\sin x\cos x}{{\cos }^{2}x+3\cos x+2}dx$

$u=\cos x\to du=\sin xdx$

$\int -\frac{u}{u^2+3u+2}du$

$-\int \frac{2}{u+2}-\frac{1}{u+1}du$

$-\int \frac{2}{u+2}du-\frac{1}{u+1}du$

$-(2\ln |\cos x+2|-\ln |\cos x+1|)$

$\int \frac{\sin x\cos x}{{\cos }^{2}x+3\cos x+2}dx=-(2\ln |\cos x+2|-\ln |\cos x+1|)+C$