Question

${\int }_0^{\frac{\pi }{2}}\frac{cosx}{(1+sinx)(2+sinx)}dx=$ [UPSEAT 1999]

• A. $log\frac{4}{3}$
• B. $log\frac{1}{3}$
• C. $log\frac{3}{4}$
• D. $Noneofthese$

Answer 1

The correct option is A $log\frac{4}{3}$

Put sin x = t $\Rightarrow$ cos x dx = dt, so reduced integral is ${\int }_0^1(\frac{1}{1+t}-\frac{1}{2+t})dt=\left[log\right(1+t)-log(2+t){]}_0^1=log\frac{2}{3}-log\frac{1}{2}=log\frac{4}{3}$