Question

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

Answer 1

$\text{let cosx=t}$

$-sinxdx=dt$

$\int \frac{sinxcox}{co{s}^{2}x+3cosx+2}dx=\int \frac{-t}{t^2+3t+2}dx=\int \frac{(t+2)-2(t+1)}{(t+1)(t+2)}dt$

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

$=ln(t+1)-2ln(t+2)+c$

$=ln\frac{t+1}{(t+2{)}^2}+c$

$=ln\frac{cosx+1}{(cosx+2{)}^2}+c$