Question

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

Answer 1

Let $\sin x=t$, then $\cos xdx=dt$
Hence,
$I={\int }_1^0\frac{dt}{(t+1)(t+2)}$

$={\int }_1^0\frac{(t+2)-(t+1)dt}{(t+1)(t+2)}$

$={\int }_1^0\frac{1}{t+1}-\frac{1}{t+2}dt$

$={\left[\ln \frac{t+1}{t+2}\right]}_1^0$

$=\left[\ln \frac{0+1}{0+2}\right]-\left[\frac{\ln }{1+1}1+2\right]$

$=\ln \left(\frac{1}{2}\right)-\ln \left(\frac{2}{3}\right)$.