Mathematics
Integration by Differentiation
Numerical Integration Through the Lens of Differentiation with an Implementation in Python
We all have a foundational understanding from high school that integration is the opposite of differentiation, as it’s the inverse operation. That’s an accurate notion. But there’s a lesser-known aspect — integration can be achieved by differentiation too. This article will provide a numerical insight into how it’s done. We will introduce a scheme that surpasses traditional direct integration methods in power because it seamlessly extends to higher-order methods and can handle any grid type.
The Idea
Let’s start by defining an integral as follows:
Keeping the upper integration limit flexible, we introduce:
Hence we can state that:
Taking the derivative simplifies to:
This is a differential equation in the function u(x). So, calculating the integral is equivalent to solving this…
