---
tag: 08Y1BBQTK9
kind: statement
role: theorem
name: Integration by parts
msc: [26A42]
uses: [0V9A43H7WV, 0XYAFCN0PT, 06SKCHDQT5]
sources:
  - url: https://en.wikipedia.org/wiki/Integration_by_parts
    locator: Product of two functions
same_as:
  wikidata: Q273328
  wikipedia: Integration by parts
---

## Statement

Let $a < b$ and let $u, v \colon [a, b] \to \mathbb{R}$ be [[continuously differentiable|06SKCHDQT5]].
Then
$$\int_a^b u(x) v'(x) \, dx = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx.$$

## Remarks

By the [[product rule|0V9A43H7WV]], $uv$ is an antiderivative of $u v' + u' v$. Apply the [[second fundamental theorem|0XYAFCN0PT]].
For indefinite integrals, the formula is $\int u \, dv = u v - \int v \, du$.
For example, $\int_0^1 x e^x \, dx = e - (e - 1) = 1$.
Mathlib states it as `intervalIntegral.integral_mul_deriv_eq_deriv_mul`.

## Explanations

### L1 human

Integration by parts is the product rule read backwards.
It moves the derivative from one factor to the other, at the cost of a boundary term.