๐Ÿ“š

All Subjects

ย >ย 

โ™พ๏ธย 

AP Calc

ย >ย 

๐Ÿ”ฅ

Unit 6

6.11 Integrating Using Integration by Parts

3 min readโ€ขapril 26, 2020

catherine

Catherine Liu


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

Was this guide helpful?

๐Ÿ” Are you ready for college apps?
Take this quiz and find out!
Start Quiz
FREE AP calc Survival Pack + Cram Chart PDF
Sign up now for instant access to 2 amazing downloads to help you get a 5
Browse Study Guides By Unit
๐Ÿ“†
Big Reviews: Finals & Exam Prep
โœ๏ธ
Free Response Questions (FRQ)
๐Ÿง
Multiple Choice Questions (MCQ)
โ™พ
Unit 10: Infinite Sequences and Series (BC Only)
๐Ÿ‘‘
Unit 1: Limits & Continuity
๐Ÿค“
Unit 2: Differentiation: Definition & Fundamental Properties
๐Ÿค™๐Ÿฝ
Unit 3: Differentiation: Composite, Implicit & Inverse Functions
๐Ÿ‘€
Unit 4: Contextual Applications of the Differentiation
โœจ
Unit 5: Analytical Applications of Differentiation
๐Ÿ’Ž
Unit 7: Differential Equations
๐Ÿถ
Unit 8: Applications of Integration
๐Ÿฆ–
Unit 9: Parametric Equations, Polar Coordinates & Vector Valued Functions (BC Only)
Join us on Discord
Thousands of students are studying with us for the AP Calculus AB/BC exam.
join now
Play this on HyperTyper
Practice your typing skills while reading Integrating Using Integration by Parts
Start Game
๐Ÿ’ช๐Ÿฝ Are you ready for the Calc AB exam?
Take this quiz for a progress check on what youโ€™ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
๐Ÿ’ช๐Ÿฝ Are you ready for the Calc BC exam?
Take this quiz for a progress check on what youโ€™ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
Hours Logo
Studying with Hours = the ultimate focus mode
Start a free study session
๐Ÿ“ฑ Stressed or struggling and need to talk to someone?
Talk to a trained counselor for free. It's 100% anonymous.
Text FIVEABLE to 741741 to get started.
ยฉ 2021 Fiveable, Inc.