How to choose u and dv in integration by parts

Integration by parts rewrites an integral as $\int u\,dv = uv - \int v\,du$. The success of the method hinges on selecting $u$ and $dv$ so that $du$ is simpler and $v$ is easy to find. A poor choice can turn a manageable problem into a more tangled one, often requiring a second application of the formula.

Heuristics for picking u

Key heuristics for selecting u and dv:

• Choose u that becomes simpler after differentiation.
• Choose dv that can be integrated without difficulty.
• Follow the LIATE order: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
• Avoid picking u that reproduces the original integral after applying the formula.

Worked example

Consider the integral $\int x e^x\,dx$. The algebraic factor $x$ simplifies to 1 when differentiated, while the exponential $e^x$ integrates to itself, making it a natural choice for $dv$. Setting $u = x$ and $dv = e^x\,dx$ leads to $du = dx$ and $v = e^x$, which satisfies the heuristics and yields a straightforward computation.

Apply the formula step by step:

1. 1Set $u = x$, so $du = dx$.
2. 2Set $dv = e^x\,dx$, so $v = e^x$.
3. 3Insert into $\int u\,dv = uv - \int v\,du$.
4. 4Compute $uv = x e^x$.
5. 5Integrate $v\,du = \int e^x\,dx = e^x$.
6. 6Combine results: $\int x e^x\,dx = x e^x - e^x + C$.

If the first choice does not simplify the integral, try swapping the roles of $u$ and $dv$ or use the LIATE hierarchy to reorder them. For instance, in $\int e^x \sin x\,dx$ choosing $u = \sin x$ (algebraic after differentiation) and $dv = e^x\,dx$ works better than the opposite assignment. Sometimes a second application of integration by parts or a reduction formula becomes necessary.

Typical u choices by function type: