Evaluate the following :
`int_(0)^(pi//2)(cosx)/((4+sinx)(3+sinx))dx`

Explanation

Let `I=int_(0)^(pi//2)(cosx)/((4+sinx)(3+sinx))dx`
Put `sin x=t" "therefore " "xdx=dt`
When `x=0, t = sin 0 =0`
When `x=(pi)/(2), t=sin.(pi)/(2)=1`
`therefore I=int_(0)^(1)(1)/((4+t)(3+t))dt`
`=int_(0)^(1)((4+t)-(3+t))/((4+t)(3+t))dt=int_(0)^(1)[(1)/(3+t)-(1)/(4+t)]dt`
`=int_(0)^(1)(1)/(3+t)dt-int_(0)^(1)(1)/(4+t)dt`
`=[log|3+t|]_(0)^(1)-[log|4+t|]_(0)^(1)`
`=(log4-log3)-(log5-log4)`
`=log 4-log 3 -log 5+log 4`
`=log((4xx4)/(3xx5))=log((16)/(15)).`