Repeated Integration by Parts
students, imagine trying to untangle a knot by pulling one loop at a time 🧵. Repeated integration by parts works the same way: when one round of integration by parts is not enough, you apply the method again and again until the integral becomes manageable. This lesson focuses on how repeated integration by parts fits inside the bigger idea of integration by parts, when to use it, and how to organize your work so the pattern becomes easier to see.
What repeated integration by parts means
Integration by parts comes from the product rule. If $u$ and $v$ are differentiable, then
$$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}.$$
Integrating both sides gives the familiar formula
$$\int u\,dv=uv-\int v\,du.$$
Repeated integration by parts means applying this formula more than once to the same integral. This usually happens when the first application still leaves an integral that is not finished, but it is simpler than the original. The method is especially useful for integrals involving products like $x^2e^x$, $x\sin x$, $x^3\cos x$, or $x^n\ln x$.
The main idea is to reduce the difficulty step by step. In many problems, each use of integration by parts lowers the power of a polynomial or turns a complicated product into a repeating pattern. That pattern is the key to solving the integral.
For example, if you want to evaluate
$$\int x^2e^x\,dx,$$
a single application of integration by parts does not finish the job. You still get an integral containing $xe^x$, which needs another application. That is repeated integration by parts in action.
How to organize the process
When students works on repeated integration by parts, organization matters. A clear structure prevents sign errors and lost terms. A good method is to keep track of each round carefully and write every step on its own line.
Start by choosing $u$ and $dv$ using the usual integration by parts strategy. A common guide is to choose $u$ so that it becomes simpler when differentiated, and choose $dv$ so that it is easy to integrate. For repeated integration by parts, $u$ is often a polynomial because repeated differentiation eventually makes it zero. Meanwhile, $dv$ is often exponential or trigonometric because those functions are easy to integrate repeatedly.
Here is the general cycle:
- Choose $u$ and $dv$.
- Compute $du$ and $v$.
- Substitute into $\int u\,dv=uv-\int v\,du$.
- If the new integral is still difficult, apply the method again.
- Continue until the remaining integral is straightforward.
A helpful habit is to label each stage. For example, if the first integral becomes $\int xe^x\,dx$, then the next round should be written as a new integration by parts problem, not as a mental shortcut. This reduces mistakes and makes your reasoning easier to follow.
Example 1: integrating $\int x^2e^x\,dx$
This is a classic repeated integration by parts problem because the power of $x$ drops each time.
Let
$$u=x^2,\quad dv=e^x\,dx.$$
Then
$$du=2x\,dx,\quad v=e^x.$$
Apply integration by parts:
$$\int x^2e^x\,dx=x^2e^x-\int 2xe^x\,dx.$$
Now the remaining integral is still not finished, so apply the method again to $\int 2xe^x\,dx$.
Let
$$u=2x,\quad dv=e^x\,dx.$$
Then
$$du=2\,dx,\quad v=e^x.$$
So
$$\int 2xe^x\,dx=2xe^x-\int 2e^x\,dx.$$
Since
$$\int 2e^x\,dx=2e^x,$$
we get
$$\int 2xe^x\,dx=2xe^x-2e^x.$$
Substitute back:
$$\int x^2e^x\,dx=x^2e^x-(2xe^x-2e^x).$$
Simplify:
$$\int x^2e^x\,dx=e^x(x^2-2x+2)+C.$$
This example shows the pattern clearly. Each round lowers the power of $x$ by $1$. Eventually, the polynomial disappears and the integral becomes easy.
Example 2: integrating $\int x\sin x\,dx$
This is another common repeated integration by parts problem, but it finishes after two steps because the polynomial has degree $1$.
Choose
$$u=x,\quad dv=\sin x\,dx.$$
Then
$$du=dx,\quad v=-\cos x.$$
Apply the formula:
$$\int x\sin x\,dx=-x\cos x+\int \cos x\,dx.$$
Now evaluate the remaining integral:
$$\int \cos x\,dx=\sin x.$$
So
$$\int x\sin x\,dx=-x\cos x+\sin x+C.$$
Even though this example only needed one full application of the formula, it still shows the repeated-steps mindset. The first use of integration by parts transforms the problem, and the remaining integral is then completed.
Example 3: integrating $\int x^2\cos x\,dx$
This example shows a stronger repeated pattern because cosine and sine switch back and forth.
Let
$$u=x^2,\quad dv=\cos x\,dx.$$
Then
$$du=2x\,dx,\quad v=\sin x.$$
So
$$\int x^2\cos x\,dx=x^2\sin x-\int 2x\sin x\,dx.$$
Now integrate $\int 2x\sin x\,dx$ by parts again.
Let
$$u=2x,\quad dv=\sin x\,dx.$$
Then
$$du=2\,dx,\quad v=-\cos x.$$
Thus
$$\int 2x\sin x\,dx=-2x\cos x+\int 2\cos x\,dx.$$
Since
$$\int 2\cos x\,dx=2\sin x,$$
we get
$$\int 2x\sin x\,dx=-2x\cos x+2\sin x.$$
Substitute this into the earlier result:
$$\int x^2\cos x\,dx=x^2\sin x-(-2x\cos x+2\sin x).$$
Simplify:
$$\int x^2\cos x\,dx=x^2\sin x+2x\cos x-2\sin x+C.$$
The alternating signs are important. Every time you integrate sine or cosine, the function changes and a negative sign may appear. students should watch carefully when distributing the minus sign from $-\int v\,du$.
A pattern for polynomial times exponential or trig functions
Repeated integration by parts is especially powerful for integrals of the form
$$\int x^n e^x\,dx,$$
$$\int x^n\sin x\,dx,$$
or
$$\int x^n\cos x\,dx.$$
The reason is that differentiating $x^n$ eventually gives $0$. That means the process cannot go on forever. Each round lowers the degree of the polynomial until the last integral is easy.
This is why the method is considered systematic, not guesswork. The work may be long, but the structure is predictable. In many textbook problems, the pattern can even be written as a table or recurrence relationship. That leads naturally to tabular integration, which is a shortcut version of repeated integration by parts.
For example, if one sets up the derivatives of $x^3$ and the repeated integrals of $e^x$, the process stops after four rows because
$$\frac{d^4}{dx^4}(x^3)=0.$$
That stopping point is what makes repeated integration by parts so effective for polynomials.
Common mistakes to avoid
Several mistakes come up often in repeated integration by parts. students can avoid many of them by checking each step carefully.
First, do not forget the negative sign in
$$\int u\,dv=uv-\int v\,du.$$
The minus sign applies to the entire new integral.
Second, make sure each new $du$ and $v$ is computed correctly. A small derivative or antiderivative error can change the entire answer.
Third, keep the structure neat. When a problem requires multiple rounds, writing the steps in separate lines helps you see the pattern and avoid missing a term.
Fourth, remember to add $C$ only at the end of an indefinite integral. During the middle of the work, you should not add a new constant after every step.
Finally, choose $u$ wisely. If $u$ does not become simpler after differentiation, repeated integration by parts may become messy instead of helpful.
How repeated integration by parts fits into integration by parts
Repeated integration by parts is not a separate rule. It is simply the same integration by parts formula used multiple times. That means it belongs directly inside the broader topic of integration by parts.
The big idea behind integration by parts is the product rule in reverse. Repeated integration by parts extends that idea to problems where the product does not disappear after one step. It is a method for handling integrals that gradually simplify through repeated differentiation and integration.
This connects to other Calculus 2 skills too. It relies on antiderivatives, algebraic simplification, and choosing the best function to differentiate. It also prepares students for tabular integration, which organizes repeated integration by parts into a faster table format.
In other words, repeated integration by parts shows the full strength of the method: a difficult product integral can often be broken into a chain of easier ones until the final answer appears.
Conclusion
Repeated integration by parts is a practical and important Calculus 2 technique. It uses the formula $\int u\,dv=uv-\int v\,du$ more than once, usually when the first application still leaves a nontrivial integral. The method works especially well for products like $x^n e^x$, $x^n\sin x$, and $x^n\cos x$ because repeated differentiation of the polynomial eventually ends. students should think of the method as a step-by-step reduction process: each round simplifies the problem a little more until it becomes manageable. Mastering this idea also builds a strong foundation for tabular integration and for the broader study of integration by parts.
Study Notes
- Repeated integration by parts means using $\int u\,dv=uv-\int v\,du$ more than once on the same integral.
- It is most useful for products such as $x^n e^x$, $x^n\sin x$, $x^n\cos x$, and $x^n\ln x$.
- Choose $u$ so that differentiating it makes it simpler, often a polynomial.
- Choose $dv$ so that it is easy to integrate, often exponential or trigonometric functions.
- Each application should reduce the complexity of the integral.
- Watch the minus sign in $uv-\int v\,du$ carefully.
- For polynomial factors, the process ends because repeated derivatives eventually give $0$.
- Repeated integration by parts is the same idea as integration by parts, just used multiple times.
- Tabular integration is an organized shortcut based on repeated integration by parts.
- Clear setup and neat writing help prevent sign errors and algebra mistakes.