Question

$\int \frac{\sin 2x}{(1+\sin x)(2+\sin x)}dx$

Answer 1

Let $I=\int \frac{sin2x}{(1+sinx)(2+sinx)}dx$

$I=\int \frac{sin2xcosx}{(1+sinx)(2+sinx)}dx$

put $sinx=t$

$cosxdx=dt$

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

$\int \frac{2tdt}{(1+t)(2+t)}=\frac{A}{(1+t)}+\frac{B}{(2+t)}$

$2t=A(2+t)+B(1+t)$

$2t=2A+At+B+Bt$

$2t=(A+B)t+2A+B$

comparing the coefficients,

$A+B=2⟹\left(1\right)$

comparing constants,

$2A+B=0⟹\left(2\right)$

Subtracting (1) from (2)

$A=-2$

$B=4$

$\frac{2t}{(1+t)(2+t)}=\frac{-2}{(1+t)}+\frac{4}{(2+t)}$

By integrating,

$\int \frac{2t}{(1+t)(2+t)}dt=-2\int \frac{dt}{(1+t)}+4\int \frac{dt}{(2+t)}=-2log|1+t|+4log|2+t|+C$

$I=-2log|1+sinx|+4log|2+t|+C$