Evaluate `int_(0)^(pi//2)(cosx)/((1+sinx)(2+sinx))dx`.

Explanation

Let sin x=1
`rArr " "cos x dx=dt`
`At x=0 " " t= sin 0=0`
`At x= .(pi)/(2), " "t=sin .(pi)/(2)=1`
`:. int_(0)^(pi//2) (cosxx)/((1+sinx )(2+sin x))dx`
` =int_(0)^(1) (1)/((1+t)(2+t))dt`
`=int_(0)^(1)((1+2)-(t+1))/((t+1)(t+2)) dt`
`=int_(0)^(1) ((1)/(t+1)-(1)/(t+2)) dt `
`=[log |t+1|-log |t+2|]_(0)^(1)`
`=(log 2-log 3) -(log 1-log 2)`
`=2 log 2-log 3`
`=log 4-log 3`
`=log .(4)/(3)`