Integration by parts

Formula

$\displaystyle \int udv$ $\,=\,$ $uv$ $-$ $\displaystyle \int vdu$

A mathematical approach of finding the integral of a product of functions from the integral of their derivative and antiderivative, is called the integration by parts.

Introduction

$f{(x)}$ and $g{(x)}$ are two functions. Take $u = f{(x)}$ and $v = g{(x)}$

The derivatives of both functions in differential form are written in the following mathematical form.

$(1) \,\,\,$ $du = f'{(x)}dx$

$(2) \,\,\,$ $dv = g'{(x)}dx$

Let’s study each part of this integral property.

$\displaystyle \int udv \,=\, \int f{(x)}.g'{(x)} dx$ and $\displaystyle \int vdu \,=\, \int g{(x)}.f'{(x)} dx$

Each equation represents the integration of a product of a function and derivative of another function, and $uv$ is a product of a function and an antiderivative of another function.

Proof

Learn how to derive the integration by parts formula in mathematical form.