Integration by parts is one of 7 integration formulas in the calculus section of this library, and it is used at ap · university level.
Why integration by parts works
Start from the product rule and integrate both sides. That says the product uv equals the sum of two integrals, and moving one of them to the other side leaves the formula. So it trades an integral you cannot do for one you hope is easier.
What each symbol means
$u$ is differentiated and $dv$ is integrated to obtain $v$.
Integration by parts: when it holds
Choose $u$ so the remaining integral becomes simpler.
When it stops applying
The method needs a piece you can actually antidifferentiate. For something like the integral of e to the x squared there is no elementary antiderivative for any sensible split, so the trade leaves you exactly where you started.
Integration by parts: a worked example
$\int xe^x dx=xe^x-\int e^x dx=e^x(x-1)+C$.
The mistake to avoid
What people do: Assigning the two parts the wrong way round, so the new integral is harder than the old one.
Why it goes wrong: In the integral of x times e to the x, letting the exponential be the part you differentiate raises the power of x instead of lowering it, and the problem gets worse each pass.
Do this instead: Choose the part that gets simpler when differentiated, which is usually a power of x or a logarithm.
Integration by parts: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $u$ is differentiated and $dv$ is integrated to obtain $v$.
- Check the conditions before substituting. Choose $u$ so the remaining integral becomes simpler.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Integration
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where integration by parts comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about integration by parts
Is there a quick way to pick the part to differentiate?
The LIATE order works for most textbook problems: logarithm, inverse trig, algebraic, trig, exponential, taking whichever comes first as the part you differentiate.
Do I need a constant at the halfway stage?
No, only at the very end. A constant attached to the antidifferentiated piece cancels itself out, so pick the simplest version available.
Can I use the method twice on one problem?
Yes, and sometimes you must. Starting with x squared times e to the x leaves an integral of x times e to the x, which needs a second pass.
What does the definite version look like?
The product term becomes a bracket evaluated at both limits, and the remaining integral keeps the same limits. The product piece turns into a number rather than staying symbolic.