Calculus › Section I · Integration techniques and improper integrals › Topic I.2
Integration by parts
Substitution cracks integrals built from a function and its derivative. But a product like x e^x — an algebraic factor glued to an exponential one — has no such chain inside. Integration by parts is the product rule run backwards: it trades one integral for another that, with a good choice, is much easier.
In this topic
- When substitution has no chain to follow
- The formula and where it comes from
- Choosing u and dv: LIATE
- Worked example: algebraic times exponential
- Worked example: the hidden product
- Repeated integration by parts
- The tabular shortcut
- When the integral comes back
- Definite integrals by parts
- Common mistakes to avoid
- Key takeaways
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Standards: 6 (BC)