Math101 · Independent Ontario learning library
Math101Integration by Parts
A rigorous, example-driven guide to integration by parts, including hypotheses, method choice, verification, and practice.
The central idea
The product rule rearranges to integration by parts: $\int u\,dv=uv-\int v\,du$. For definite integrals, evaluate $uv$ at both bounds and subtract the remaining definite integral. The method is useful when differentiating one factor simplifies it while the other factor is readily integrated.
Definitions, hypotheses, and notation
A useful preference list—logarithmic, inverse trigonometric, algebraic, trigonometric, exponential—can suggest $u$, but it is a heuristic, not a theorem. The real test is whether $du$ simplifies and $dv$ integrates. For polynomial times exponential or sine, repeated integration by parts eventually differentiates the polynomial to zero.
For integrals such as $\int e^x\cos xdx$, applying parts twice reproduces the original integral. Collecting that term algebraically then solves the equation. Constants of integration should be added only after the indefinite equation is resolved.
Conceptual meaning
Integration by parts transfers a derivative from one factor to another. It does not eliminate complexity automatically; the choices of $u$ and $dv$ should make the new integral simpler or create an equation involving the original integral.
A dependable method and decision rule
- Factor the integrand conceptually into a choice of $u$ and $dv$.
- Prefer a $u$ that simplifies under differentiation and a $dv$ with known antiderivative.
- Compute $du$ and $v$ explicitly.
- Substitute into $uv-\int vdu$ with the minus sign visible.
- Repeat, solve algebraically, or apply bounds as the resulting structure requires.
