Integration by Parts — Formula & Examples

Integration by parts helps you integrate products like $x e^x$ or $x \ln x$ when one factor gets simpler after differentiation. The goal is not to use a fancy formula for its own sake. The goal is to turn the original integral into an easier one.

It comes from reversing the product rule. If the new integral is not simpler, integration by parts is probably the wrong move.

Integration By Parts Formula

If you choose a function $u$ and a differential part $dv$, then

$$\int u\,dv = uv - \int v\,du$$

This is the integration by parts formula. It is useful only when the new integral $\int v\,du$ is easier than the original one.

Why The Formula Works

Start with the product rule written in differential form:

$$d(uv) = u\,dv + v\,du$$

Integrate both sides with respect to $x$:

$$\int d(uv) = \int u\,dv + \int v\,du$$

So

$$uv = \int u\,dv + \int v\,du$$

and rearranging gives

$$\int u\,dv = uv - \int v\,du$$

You do not need to re-derive it every time, but this is why the minus sign is there.

How To Choose $u$ And $dv$

Choose $u$ as the part that becomes simpler after differentiation. Choose $dv$ as the part you can integrate without much trouble.

One common heuristic is LIATE: logarithmic, inverse trig, algebraic, trig, exponential. It is only a guide, not a rule, but it often helps when more than one choice seems reasonable.

In practice, integration by parts is common when you see:

• a polynomial times $e^x$ or a trig function,
• a logarithm such as $\ln x$, often treated as $\ln x \cdot 1$,
• an inverse trig function such as $\arctan x$.

The best quick check is this: after you pick $u$, ask whether $du$ is clearly simpler. If the answer is no, try a different choice.

Worked Example: $\int x \ln x\,dx$

This is a standard example because $\ln x$ becomes much simpler when you differentiate it. Rewrite the integrand as a product:

$$\int x \ln x\,dx = \int (\ln x)(x)\,dx$$

The condition matters here: $\ln x$ is defined for $x > 0$, so we work on that domain.

Choose

$$u = \ln x \qquad dv = x\,dx$$

Then

$$du = \frac{1}{x}\,dx \qquad v = \frac{x^2}{2}$$

Apply the formula:

$$\int x \ln x\,dx = \frac{x^2}{2}\ln x - \int \frac{x^2}{2}\cdot \frac{1}{x}\,dx$$

Simplify the remaining integral:

$$\int x \ln x\,dx = \frac{x^2}{2}\ln x - \frac{1}{2}\int x\,dx$$

Then integrate:

$$\int x \ln x\,dx = \frac{x^2}{2}\ln x - \frac{1}{2}\cdot \frac{x^2}{2} + C$$

So the final answer is

$$\int x \ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C$$

Differentiate the result to check it:

$$\frac{d}{dx}\left(\frac{x^2}{2}\ln x - \frac{x^2}{4}\right) = x\ln x$$

That check is the fastest way to catch sign errors.

Common Integration By Parts Mistakes

1. Choosing $u$ and $dv$ so that the new integral is harder than the original one.
2. Forgetting the minus sign in $uv - \int v\,du$.
3. Differentiating $u$ correctly but integrating $dv$ incorrectly.
4. Forgetting that some expressions, like $\ln x$, come with domain conditions.
5. Assuming every product should use integration by parts. Sometimes substitution or a basic rule is better.

When Integration By Parts Is Useful

Use this method when the integrand has structure that improves after one differentiation step. Typical cases include:

• polynomial times exponential, such as $\int x e^x\,dx$,
• polynomial times trig, such as $\int x \cos x\,dx$,
• logarithms or inverse trig functions multiplied by $1$ or another simple factor.

If the method does not simplify the integral, stop and reassess. Integration by parts is useful because it reduces complexity, not because the formula applies mechanically.

Try A Similar Problem

Try

$$\int x \sin x\,dx$$

Use the same decision process: choose the part that simplifies when differentiated, apply the formula once, and then differentiate your answer to verify it.