Integration by Parts

Integrates products of functions; the reverse of the product rule.

\int u d v = uv - \int v d u

Variables

u: Part to differentiate
dv: Part to integrate

Explanation

Choose $u$ using LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) — the earlier in the list, the better a choice for $u$ .

Example

$\int x e^{x} d x$ : $u = x$ , $d v = e^{x} d x$ gives $x e^{x} - e^{x} + C$ .

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Related formulas

Fundamental Theorem of Calculus

Connects definite integrals to antiderivatives.