`int_(0)^(pi//2)(cosx)/((1+sinx)(2+sinx))dx` is equal to

Options

A. `"log"(4)/(3)`
B. `"log"(1)/(3)`
C. `"log"(3)/(4)`
D. None of these

Answer

A

Explanation

Let `l=int_(0)^(pi//2)(cosx)/((1+sinx)(2+sinx))dx`
Put sin `x=timpliescosxdx=dt`
`thereforel=int_(0)^(1)(dt)/((1+t)(2+t))`
`=int_(0)^(1)(1)/((1+t))dt-int_(0)^(1)(1)/((2+t))dt`
`[log(1+t)-log(2+t)]_(0)^(1)`
`=(log2-log3)-(0-log2)`
`="log"(4)/(3)`