Question

Solve :
$I=\int \frac{\cos x}{2{\sin }^{2}x+3\sin x+1}dx$

Answer 1

$I=\int \frac{\cos xdx}{2{\sin }^{2}x+3\sin x+1}$

$\sin x=t$

$\Rightarrow \cos xdx=dt$

$I=\int \frac{dt}{2t^2+3t+1}$

$=\int \frac{dt}{(2t+1)(t+1)}$

$=\int \frac{2(t+1)-(2t+1)}{(2t+1)(t+1)}dt$

$=\int \frac{2}{2t+1}dt-\int \frac{1}{t+1}dt$

$=\int \frac{2}{2}ln(2t+1)-ln(t+1)$

$=ln\left(\frac{2t+1}{t+1}\right)+c$ Answer.