`int_(0)^(pi//2)(cosx)/((1+sinx)(2+sinx))dx` का मान ज्ञात कीजिए ।

Explanation

माना `sinx=t`
`rArr" "cos xdx=dt`
`{:(x=0 ," पर ",t=sin0=0),(x=(pi)/(2)," पर ",t=sin.(pi)/(2)=1):}`
`therefore int_(0)^(pi//2)(cosx)/((1+sinx)(2+sinx))dx`
`" "=int_(0)^(1)(1)/((1+t)(2+t))dt`
`" "=int_(0)^(1)((t+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)`
`=(log2-log3)-(log1-log2)`
`=2log2-log3`
`=log4-log3`
`=log.(4)/(3)`