Question

${\int }_0^{\pi /2}\frac{\cos xdx}{(1+\sin x)(2+\sin x)}$

Answer 1

$\sin x=\Rightarrow \cos x.dx=dt$

${\int }_0^{\pi /2}\frac{\cos x.dx}{(1+\sin x)(2+\sin x)}={\int }_0^{\pi /2}\frac{1}{(t+1)(t+2)}dt$

${\int }_0^{\pi /2}\frac{1}{t+1}dt-{\int }_0^{\pi /2}\frac{1}{t+2}dt=\left[ln\right(t+1)-ln(t+2)+c{]}_0^{\pi /2}$

$=\left[ln\right(\sin x+1)-ln(\sin x+2){]}_0^{\pi /2}$

$=[ln2-ln3-ln1+ln2]=2ln2-ln3-ln1.$