# How do I integrate x * ln x

## Derivation of the antiderivative of the natural logarithm

The complete derivation of the antiderivative of the natural logarithm with step-by-step explanation.

### Explanation

1. Find the integral of the natural logarithm function ln (x)
2. Partial integration, also called product integration, is used for integration. As the name implies, product integration requires a product that can be integrated. Here we use a trick: we multiply the integrand times 1, which does not change it, but which at the same time creates a product that we can integrate.
3. In the case of partial integration, the choice of f (x) and g '(x) is important (see also the article on partial integration), since a wrong choice increases the workload considerably. We choose g '(x) = 1 and f (x) = ln (x).
4. We now have to integrate g '(x), while we have to derive f (x). For both functions, their respective antiderivative or derivative can be easily determined.
5. Next we insert the calculated antiderivatives or derivatives of f (x) and g (x) into the formula for the partial integration.
6. There is another integral that has yet to be solved. The Integrad is shortened from x/x to 1, and can thus be easily integrated.
7. The integral has now been calculated and completes the formula for partial integration from (5).