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Math · Calculus II · Concept

Integration by parts: formula and examples

Integration by parts is the product rule run backward: the integral of u dv equals uv minus the integral of v du. It trades an integral you cannot do for one you can, when the integrand is a product in which one factor gets simpler when differentiated, such as x, and the other is easy to integrate, such as eˣ or cos x. It even integrates ln x, by treating it as ln x times 1.

Where the formula comes from

The product rule says (uv)′ = u′v + uv′. Integrate both sides and rearrange, writing du for u′ dx and dv for v′ dx.

∫ud⁢v=u⁢v−∫vd⁢u

Choosing u and dv

Pick u to be the factor that gets simpler when you differentiate it, and dv to be something you can integrate. A common guide, remembered as LIATE, is the order logarithmic, inverse trigonometric, algebraic, trigonometric, exponential: a factor earlier in the list usually makes the better u.

Definite integrals

For a definite integral from a to b, evaluate uv at both limits and subtract, then integrate v du over the same limits.

∫abud⁢v=[u⁢v]ab−∫abvd⁢u

Using parts more than once

Sometimes the new integral needs parts again, as for x²eˣ: each round lowers the power of x by one. A table of u’s derivatives beside repeated integrals of dv, the tabular method, keeps the rounds organized. For eˣ sin x the original integral comes back after two rounds; move it to the other side and solve for it.

Check by differentiating

Differentiate the answer. The product rule should return the integrand exactly.

Common mistakes

  • Choosing u = eˣ and dv = x dx for the integral of xeˣ: the new integral, of (x²/2)eˣ, is harder than the original.
  • Dropping the minus sign in uv − ∫v du.
  • Evaluating uv at only one limit in a definite integral.
  • Forgetting the + C.

Key terms

Integration by parts
A method that reverses the product rule: ∫u dv = uv − ∫v du. It suits a product in which one factor gets simpler when differentiated and the other is easy to integrate.
Antiderivative
A function whose derivative is the given function, such as x³ for 3x². Any two antiderivatives on one interval differ by a constant, which is why answers carry + C.
Integration by substitution
A method that reverses the chain rule: when an integrand has the form f(g(x))·g′(x), set u = g(x), so du = g′(x) dx, and integrate f(u) instead. A definite integral also converts its limits to u-values.

Work through an example

Find the integral of xeˣ with respect to x.

Integrate xeˣ by parts →

Integrate ln x by parts →

Integrate by parts twice →

Sources and scope

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Make it concrete

Try in the workspace

Open the example inputs, change a value and keep a useful result on your board.

Check it in Derivative & antiderivative checks Check it in Math Open worked example on a board Product rule in Math Reference

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