Question

What is the integral of `ln(x)/x`?

Answer 1

Lets start by breaking down the function.

`(ln(x))/x = 1/x ln(x)`

So we have the two functions;

`f(x) = 1/x`
`g(x) = ln(x)`

But the derivative of `ln(x)` is `1/x`, so `f(x) = g'(x)`. This means we can use substitution to solve the original equation.

Let `u = ln(x)`.

`(du)/(dx) = 1/x`

`du = 1/x dx`

Now we can make some substitutions to the original integral.

`int ln(x) (1/x dx) = int u du = 1/2 u^2 + C`

Re-substituting for `u` gives us;

`1/2 ln(x)^2 +C`