We will provide examples of how you solve integrals numerically in Python. Let’s recall from statistics that the mean value can be calculated as.

\( E(X) = \frac{1}{b-a} \int_{a}^{b}f(x)dx\)

\((b-a) E(X) = \int_{a}^{b} f(x) dx\)

\((b-a)\frac{1} {N}\sum_{i}f(x_i) \approx \int_{a}^{b}f(x)dx \)

This implies that we can find an approximation of an interval by calculating the average value times the range that we intergate.

## Example of Monte Carlo Integration

Let’s say that we want to calculate the following integral where from WolframAlpha we get the solution:

\(\int_{5}^{20}\frac{x}{(x+1)^3}dx = \frac{125}{1176}\approx 0.10629\)

**Solution with Python**

import numpy as np Ν = 100000000 a = 5 b = 20 x = np.random.uniform(a,b,Ν) f_x = x/((1+x)**3) print(np.mean(f_x)*(b-a))

`0.10629043477066367`

**Not bad! The Month Carlo Integration returned a very good approximation!**