Question

How do I find the integral `int(x*ln(x))dx` ?

Answer 1

We will use integration by parts.

Remember the IBP's formula, which is

`int u dv = uv - int v du`

Let `u = ln x`, and `dv = x dx`. We have chosen these values because we know that the derivative of `ln x` is equal to `1/x`, meaning that instead of integrating something complex (a natural logarithm) we now will end up integrating something pretty easy. (a polynomial)

Thus, `du = 1/x dx`, and `v = x^2 / 2`.

Plugging into the IBP's formula gives us:

`int x ln x dx = (x^2 ln x)/2 - int x^2 / (2x) dx`

An `x` will cancel off from the new integrand:

`int x ln x dx = (x^2 ln x)/2 - int x / 2 dx`

The solution is now easily found using the power rule. Don't forget the constant of integration:

`int x ln x dx = (x^2 ln x)/2 - x^2 / 4 + C`