Integration by Parts

Summary

Integration by parts is the integral form of the product rule. Choose factors $u$ and $d v$ so that $\int v d u$ is easier than $\int u d v$ .

Prerequisites

Integrals, Derivatives, product rule

Formula

\int u d v = u v - \int v d u .

For definite integrals,

\int_{a}^{b} u d v = [u v]_{a}^{b} - \int_{a}^{b} v d u .

Conditions / Assumptions

$u$ and $v$ should be continuously differentiable on the interval of integration (sufficient for elementary use).
Strategy (LIATE/ILATE heuristics): prefer $u$ to be a log, inverse trig, algebraic, trig, or exponential factor in that rough order—but check that $d v$ is easy to integrate.

Worked Example

Classic parts: $\int x e^{x} d x$

Let $u = x$ , $d v = e^{x} d x$ , so $d u = d x$ , $v = e^{x}$ . Then

\int x e^{x} d x = x e^{x} - \int e^{x} d x = x e^{x} - e^{x} + C = e^{x} (x - 1) + C .

Substitution, not parts: $\int x^{2} e^{x^{3}} d x$

Here the derivative of the exponent is proportional to $x^{2}$ . Set $w = x^{3}$ , $d w = 3 x^{2} d x$ :

\int x^{2} e^{x^{3}} d x = \frac{1}{3} \int e^{w} d w = \frac{1}{3} e^{x^{3}} + C .

Parts is the wrong primary tool for this integral.

Common Mistakes

Writing a broken formula that confuses antiderivatives of $u$ and $v$ with $u$ and $d v$ .
Using parts on a pure substitution problem such as $\int x^{2} e^{x^{3}} d x$ and producing algebraically incorrect reductions.

Connections

Related techniques: Integrals, Derivatives, substitution, and reduction formulas
Series: Infinite Series and term-by-term integration

References

Integration by parts is in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Section 3.1, https://openstax.org/details/books/calculus-volume-2 ↩︎