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.



  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).