Join Our Discord

Learn Moreย ย ๐Ÿ‘‡

caret down
caret up
discord-logo

Join our Discord

1,000s of Fiveable Community students are already finding study help, meeting new friends, and sharing tons of opportunities among other students around the world! Don't miss out!ย  ๐ŸŽ‰

group of students
group of students

โ™พ๏ธย ap calc

ย ย >ย ย 

๐Ÿ”ฅย Unit 6

ย ย โ€ขย ย โฑ๏ธ3 min read

6.11 Integrating Using Integration by Parts

Catherine Liu

catherine liu

โฑ๏ธ April 26, 2020

๐Ÿ“…โŒจ๏ธ

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(565).png?alt=media&token=686387f8-43f9-45fc-b1c8-f5f73b89a3b2

AP Description & Expectations ๐ŸŽฏ

  • Enduring Understanding FUN-6: Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.ย 

  • Essential Knowledge FUN-6.E.1: Integration by parts is a technique for finding antiderivatives.

What is Integration by Parts?

Integration by parts is a method of integration that transforms an integral of a product of functions into an integral of the product of one functionโ€™s derivative and the others antiderivative. Although this sounds confusing at first, we can build this concept from the product rule:ย ๐Ÿฅ‡

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(567).png?alt=media&token=b06cf3f5-4c4f-4e3a-8a40-9022233cf58c

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(568).png?alt=media&token=d287c198-2a96-4edf-b3e1-275d536b2d2e

Whenever thereโ€™s an integral where the integrand (the thing being integrated) is a product of two functions, one of the functions will be f(x) and the other will be gโ€™(x). If other methods like u-substitution donโ€™t work, try integration by parts. ๐Ÿ’ฏ

Another Form of the Equation โ“

Sometimes, you might see the equation written like this:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(570).png?alt=media&token=6f151ba6-f105-40fc-a261-c613a7521cac

This is the same concept in a different form.

One of the functions will be U and the other will be V. However, we still need to know what dV and dU actually mean. If U is a function of x, then U = f(x). Taking the derivative of both sides, we find that dU = fโ€™(x)dx. Same thing for V: If V = g(x),ย  then dV = gโ€™(x)dx. This new way of writing the integration by parts formula is just a sneakier version of the original formula. ๐Ÿคญ

How and Why Would I Use This Formula? ๐Ÿค”

The best way to explain this formula is through an example. Take the following integral:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(572).png?alt=media&token=4f6238b4-8451-4825-b925-ebbd874acd54

At first glance, you may try u-substitution, but this method will take you nowhere. Instead, notice that the integrand is a product of two functions. This is an opportunity to use integration by parts. Letโ€™s recall the formula:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(574).png?alt=media&token=328ac1fe-27e1-480c-8b04-ede802dcb4a9

The first step should be to figure out what f(x) and gโ€™(x) should be. This step is probably the most challenging of this method, because you have to choose which is which in order to make the problem work.ย 

This is where a handy acronym comes into play: L.I.A.T.E. It stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential.

The function closer to the โ€œEโ€ side should be gโ€™(x) and the function closer to the โ€œLโ€ side should be f(x).ย 

For our example problem, that means:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(576).png?alt=media&token=40b70a38-9e35-421d-b317-64cbe13e0913

We know we need g(x) and fโ€™(x) to use the integration by parts formula, so letโ€™s find those now:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(579).png?alt=media&token=42a92b30-db94-477f-8991-8a65ea126268

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(580).png?alt=media&token=3b46d1ec-65d0-48fd-92df-94fee14a8049

Steps to Take for Integration by Parts

  1. Check to make sure that u-substitution and other methods donโ€™t work.

  2. Check to make sure that the integrand is a product of two functions.

  3. Use L.I.A.T.E. to pick which function should be f(x) and which should be gโ€™(x).

  4. Plug into the formula and integrate.

Note: You may have to use integration by parts more than once if your resulting integral is also a product of two functions that can't be solved using u-substitution.ย 

Worked Examples ๐Ÿ“”

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(582).png?alt=media&token=af88a834-b291-45b1-a007-2e2a9a560b21

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(589).png?alt=media&token=92b8a379-49a7-4a34-ae3e-005ce13d9c12

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(590).png?alt=media&token=7b3d66de-0f3a-462b-b9f3-b5795d072612


https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(592).png?alt=media&token=ea38b513-609e-4286-94fe-8a01a00045b0

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(594).png?alt=media&token=9f13a11e-8693-4bc9-9a61-41c6b5eac949

A tent in the woods with a fire

Get your ap calc survival pack!

Download our ap calc survival pack and get access to every resource you need to get a 5.

ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

Fiveable

Join Our Discord

discord-logo

Fiveable Community students are already meeting new friends, starting study groups, and sharing tons of opportunities for other high schoolers. Soon the Fiveable Community will be on a totally new platform where you can share, save, and organize your learning links and lead study groups among other students!๐ŸŽ‰

*apยฎ and advanced placementยฎ are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.

ยฉ fiveable 2021 | all rights reserved.

2550 north lake drive
suite 2
milwaukee, wi 53211