Integration by Parts (LIATE) Reference Cheat Sheet

Key Facts

• The integration by parts formula is $\int u\,dv = uv - \int v\,du$.
• The definite integral form is $\int_a^b u\,dv = \left[uv\right]_a^b - \int_a^b v\,du$.
• LIATE ranks choices for $u$ as logarithmic, inverse trigonometric, algebraic, trigonometric, then exponential.
• After choosing $u$, compute $du$ by differentiating and compute $v$ by integrating $dv$.
• Choose $u$ so that $du$ is simpler than $u$, such as choosing $u = x^3$ because $du = 3x^2\,dx$.
• For $\int x e^x\,dx$, choose $u = x$ and $dv = e^x\,dx$ to get $\int x e^x\,dx = x e^x - e^x + C$.
• For $\int \ln x\,dx$, use $u = \ln x$ and $dv = dx$ to get $\int \ln x\,dx = x\ln x - x + C$.
• In tabular integration, multiply diagonally with alternating signs $+$, $-$, $+$, $-$ until the derivative column ends or repeats.

Vocabulary

Integration by Parts

A technique for integrating products that rewrites $\int u\,dv$ as $uv - \int v\,du$.

LIATE

A guideline for choosing $u$ in integration by parts, ordered as logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential.

Tabular Method

A shortcut for repeated integration by parts that organizes derivatives, integrals, and alternating signs in a table.

Repeated Integration by Parts

Using integration by parts more than once in the same problem, usually because the remaining integral is still a product.

Differential

A notation such as $du$ or $dv$ that represents the derivative part used to track substitution and integration steps.

Antiderivative

A function $F(x)$ whose derivative is the original function, so $F'(x) = f(x)$.