# Numerical integration of a function

For a particular physical system, the force is given by $$F(r)=-\frac{d\Phi(r)}{dr}=\frac{r^5+2ar^3+a^2r-4a}{r^4(r^3+ar+2a)}$$ where $$\Phi(r)$$ is the potential energy which can be obtained as the integral of $$F(r)$$. However, the integral is very complicated to solve as the cubic polynomial in the denominator of the above equation has complex roots (refer this question in Math.SE). Actually I need to obtain the variation of $$\Phi(r)$$ as a function of $$r$$, i.e., the $$\Phi(r)-r$$ plot.

As the evaluation of the integral is complicated, I am trying to obtain the integral curves of $$F(r)$$ using the dataset of the $$F(r)-r$$ plot. In this sense, I need to integrate a dataset. However, I am not sure whether this is possible.

Question: Is it possible to integrate a dataset to obtain the integral curve of a function?

• Google for Gauss quadrature, or Clenshaw Curtis. I quite like the latter. – gnasher729 Nov 22 '19 at 11:21