Integration by parts: choosing what to differentiate

Understand the idea

Integration by parts reverses the product rule. It is useful when an integrand contains a product whose factors play different roles: differentiating one factor simplifies it, while the other has an antiderivative you can find. The goal is to replace the original integral with an easier one, not simply to make it look different.

Choose $\displaystyle u$ and $\displaystyle dv$ together. A polynomial often becomes simpler after differentiation, while exponential and trigonometric factors are usually manageable to integrate. Logarithms can be good choices for $\displaystyle u$ even when the original integrand does not visibly look like a product: write $\displaystyle \ln x=(\ln x)\cdot1$.

For definite integrals, evaluate the product at both bounds and keep the remaining integral on the same interval. For an indefinite integral, add one arbitrary constant at the end. If the remaining integral is harder, reconsider the choice of $\displaystyle u$ before continuing.

Worked example

Find $\displaystyle \int x e^{2x}\,dx$.

1. Take $\displaystyle u=x$ and $\displaystyle dv=e^{2x}\,dx$. Then $\displaystyle du=dx$ and $\displaystyle v=\tfrac12e^{2x}$.
2. Integration by parts gives $\displaystyle \tfrac12xe^{2x}-\tfrac12\int e^{2x}\,dx$.
3. The remaining integral contributes $\displaystyle \tfrac14e^{2x}$, which must be subtracted.
4. Differentiating the resulting expression produces $\displaystyle xe^{2x}$; the extra exponential terms cancel.