Question

`intlogx/(1 + logx)^2dx` = ______ 

Options

• `1/(1 + logx) + c`

• `x/(1 + logx)^2 + c`

• `x/(1 + logx) + c`

• `1/(1 + logx)^2 + c`

Answer 1

`intlogx/(1 + logx)^2dx` = `underline(x/(1 + logx) + c)`

Explanation:

Put logx = t ⇒ x = e^t ⇒ dx = e^t dt

∴ `intlogx/(1 + logx)^2dx = intt/(1 + t)^2 e^t  dt`

= `inte^t [(t + 1 - 1)/(1 + t)^2]dt`

= `inte^t [1/(1 + t) - 1/(1 + t)^2]dt`

= `e^t/(1 + t) + c`

= `x/(1 + logx)` + c