Integration by parts

Integration by parts is the integration analogue of the product rule. It is useful when an integrand is a product and differentiating one factor simplifies it.

Start from the product rule:

$$\frac{d}{dx}(uv)=u^{\prime }v+u{v}^{\prime }.$$

Rearrange and integrate:

$$\int udv=uv-\int vdu.$$

Here $u=u(x)$ is the part we choose to differentiate, and $dv=v^{\prime }(x)dx$ is the part we choose to integrate.

How to choose $u$

A practical heuristic is LIATE: logarithmic, inverse trig, algebraic, trigonometric, exponential. Pick $u$ from the earliest available type, because it usually becomes simpler under differentiation. This is a heuristic, not a theorem.

Worked examples

For $\int x{e}^{x}dx$, choose $u=x$ and $dv=e^{x}dx$. Then $du=dx$ and $v=e^x$:

$$\int x{e}^{x}dx=x{e}^x-\int e^{x}dx=e^x(x-1)+C.$$

For $\int \ln xdx$, treat it as $\int 1\cdot \ln xdx$. Choose $u=\ln x$, $dv=dx$, so $du=\frac{1}{x}dx$ and $v=x$:

$$\int \ln xdx=x\ln x-\int 1dx=x\ln x-x+C.$$

Definite version

$${\int }_a^{b}udv={\left[uv\right]}_a^b-{\int }_a^{b}vdu.$$

This is common in physics when moving derivatives between factors, for example in variational calculations and energy methods. It is also the one-dimensional ancestor of integration-by-parts identities used with the Del operator ∇ in Vector Calculus index.

Common pitfalls

• Forgetting the minus sign in $uv-\int vdu$.
• Choosing $u$ so that $du$ becomes more complicated.
• Dropping the constant $C$ for indefinite integrals.
• Mixing up $dv$ and $v$: $dv$ includes the differential, while $v$ is its antiderivative.

Quick diagnostic

If repeated integration by parts returns a multiple of the original integral, move that term to the left and solve algebraically. This happens for examples like $\int e^x\sin xdx$.