Evaluate the following integrals:
`int(logx)/((1+logx)^(2))dx`

Explanation

Let `" "I=int(logx)/((1+logx)^(2))dx`
Putting `logx=t rArr x = e^(x)`
`rArr dx= e^(t)dt`
`therefore" "I=int(t)/((1+t)^(2)).e^(t)dt`
`=inte^(y)(((1+t)-1)/((1+t)^(2)))` ltBrgt `=inte^(t)((1)/(1+t)-(1)/((1+t)^(2)))dt" ...(i)"`
`"Let "f(t)=(1)/(1+t)`
`rArr" "f'(t)=(-1)/((1+t)^(2))`
`therefore" "I=inte^(t)[f(t)+f'(t)]dt`
`=e^(t)(t)+c`
`=e^(t)(1)/(1+t)+c=(x)/(1+logx)+c`