How do i do this equation, with what technique ?

Question

Answer 1

∫ ln x / ( x + x ln) dx = ∫ ln x / x(1 + ln x) dx

Let u = ln x , du = 1/x dx

∫ ln x / x(1 + ln x) dx = ∫ u / ( 1+u) du

= ∫( u +1 - 1) / ( 1+u) du , adding 1 and subtracting 1 from the numerator

= ∫ u + 1 / (1+u) - 1/ ( 1+u) du

= ∫ 1 - 1/ ( 1+u) du

= u - ln (1+u)

= ln x - ln ( 1+ ln x )

I hope all the steps are clear . if some thing not clear you can ask.