Main content

### Course: AP®︎/College Calculus BC > Unit 6

Lesson 13: Using integration by parts- Integration by parts intro
- Integration by parts: ∫x⋅cos(x)dx
- Integration by parts: ∫ln(x)dx
- Integration by parts: ∫x²⋅𝑒ˣdx
- Integration by parts: ∫𝑒ˣ⋅cos(x)dx
- Integration by parts
- Integration by parts: definite integrals
- Integration by parts: definite integrals
- Integration by parts challenge
- Integration by parts review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Integration by parts review

Review your integration by parts skills.

## What is integration by parts?

Integration by parts is a method to find integrals of products:

or more compactly:

We can use this method, which can be considered as the "reverse product rule," by considering one of the two factors as the derivative of another function.

*Want to learn more about integration by parts? Check out this video.*

## Practice set 1: Integration by parts of indefinite integrals

Let's find, for example, the indefinite integral $\int x\mathrm{cos}x{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ . To do that, we let $u=x$ and $dv=\mathrm{cos}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ :

Now we integrate by parts!

Remember you can always check your work by differentiating your result!

*Want to try more problems like this? Check out this exercise.*

## Practice set 2: Integration by parts of definite integrals

Let's find, for example, the definite integral ${\int}_{0}^{5}x{e}^{-x}dx$ . To do that, we let $u=x$ and $dv={e}^{-x}{\textstyle \phantom{\rule{0.167em}{0ex}}}dx$ :

Now we integrate by parts:

*Want to try more problems like this? Check out this exercise.*

## Want to join the conversation?

- Thanks Sal. I am the master of integration by parts. Do not test me. I am now the best, I think I should do the testing.(37 votes)
- in the int (0 -> pi) of xsin(2x)dx problem, in the solution, the third to last line, shouldn't that be (sin(2x)/4) not (sin(4x)/4)? or am I missing something?(11 votes)
- You are correct, it is a typo, though it does not effect the result since sin(nπ) = 0 for all integers n.

You can, and should, report the error since you found it.(8 votes)

- Why hasn't Sal explained about the compact form of Integration by parts??i don't understand it!! It contradicts to what Sal said about differentials earlier that the differentials are not numbers or function which can't cancelled or algebraically manipulated!!(10 votes)
- The "compact form" is just a different way to write the form used in the videos. Basically, the only difference is that the "video form" uses prime notation (f'(x)), and the "compact form" uses Leibniz notation (dy/dx). If you are used to the prime notation form for integration by parts, a good way to learn Leibniz form is to set up the problem in the prime form, then do the substitutions f(x) = u, g'(x)dx = dv, f'(x) = v, g(x)dx = du. At least, that's how it clicked for me.

As far as the manipulating differentials goes, it's true that you can't just treat differentials like they are normal terms in an equation (as if dx were the variable d times the variable x), but it is legal to split up the dy/dx when differentiating both sides of an equation. The concept here is exactly the same as what is used when doing u-substitution (URL to video below if you need it).

https://www.khanacademy.org/math/calculus-home/integration-techniques-calc/u-substitution-calc/v/u-substitution

Hope this helps, and good luck with your work!(5 votes)

- Like Bhoovesh I am also fuzzy about the compact notation. It seems that the confusion is not with Leibniz notation vs Newton's, but rather I am concerned about falling in a pit as a consequence of having only one letter in an expression for which I am accustomed to two. The dropping of the x's and dx's makes me nervous about getting fouled up as a consequence of x not being the only variable. Where is y, and how does one keep track of it with the more compact notation? I would like to know the conventions and rules for this.(7 votes)
- For a moment, consider the product rule of differentiation. Sal writes (in the intro video)
`d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)`

Now, are you used to this notation? :`d/dx[f(x)] = f'(x)`

I hope so. Now, since both are functions of x, for short form notation we can leave out the x.

So

becomes`d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)`

`(fg)' = f'g + fg'`

Same deal with this short form notation for integration by parts.

This article talks about the development of integration by parts:

http://www.sosmath.com/calculus/integration/byparts/byparts.html

This one a bit deeper:

https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/intbypartsdirectory/IntByParts.html

These articles really just serve to confirm the ubiquity of the short form notation*and*they may help you get you more comfortable with it:

http://www.mathsisfun.com/calculus/integration-by-parts.html

https://en.wikipedia.org/wiki/Integration_by_parts

http://mathworld.wolfram.com/IntegrationbyParts.html

http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx

You may want to suggest to the Khan site to make a video talking about the the conversion and utility of the long form to short form notation. I suspect however, with more practice, exposure and careful consideration, you will get it on your own.

I hope this helped a bit.

Keep Studying(8 votes)

- Hi. There is a gap of explanation about the compact notation. I didn't catch why 'dx' becomes 'dv' and 'du'. Thank you.(7 votes)
- In case anyone else is confused,

to find 'dv' from 'v', take the derivative and multiply by 'dx'. You can think of this as because the derivative of 'v' is 'dv/dx', so given`dv/dx = a => dv = adx`

. This is similar to one way of doing u-substitution.(3 votes)

- Why does the integral of e^5x dx = 1/5 e^5x? Is it an application of the reverse chain rule?

Thanks very much!(6 votes)- That's one way of thinking about it, yes. As you continue on in math, this will become almost second-nature and you won't even think about the chain rule when integrating simple exponential functions.(3 votes)

- What is the use of integration? When do we use it in our practical lives?(0 votes)
- we can use integration to find Displacement from Velocity, and Velocity from Acceleration, Voltage across a Capacitor, Moments of Inertia by Integration and many more...(14 votes)

- How would you integrate something in parentheses, like (x^2 +1)^1/2?(3 votes)
- That depends hugely on what's in the parentheses. √(x²+1) can be done by trigonometric substitution, but √(x³+1) cannot be done by elementary means.(3 votes)

- ILATE technique will be useful(3 votes)
- Is there a reverse division rule that can sometimes serve as a substitute for this? An example where this would be useful is
*(ln(5x))/(x^2)*(2 votes)- the quotient rule for derivatives is just a special case of the product rule. f(x)/g(x) = f(x)*(g(x))^(-1) or in other words f or x divided by g of x equals f or x times g or x to the negative one power. so it becomes a product rule then a chain rule.

So when you have two functions being divided you would use integration by parts likely, or perhaps u sub depending.

Really though it all depends. finding the derivative of one function may need the chain rule, but the next one would only need the power rule or something. It's kinda hard to predict if two functions being divided need integration by parts or what to integrate them.(2 votes)