Question

What is the integral of `int ln x / x dx`?

Answer 1

`(1/2)(ln |x|)^2+c`

Explanation:

Integration by parts, with `u=ln x`, `v=ln x`, `(du)/(dx)=1/x`, `(dv)/(dx)=1/x`

`I=int(1/x)ln x dx`
`=(ln x)(ln x)-int(ln x).(1/x)dx`
`=(ln x)(ln x)-I`
So
`2I= ln(x)^2`
`I=(1/2)(ln x)^2+c`

Alternatively, by inspection, the integral is of the form `f'(x)f(x)` with `f(x)=ln(x)`, `f'(x)=1/x`. The integral of `f'(x)f(x)` is `(1/2)(f(x))^2`