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2 min read

6.12 Using Linear Partial Fractions

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=15ee1323-7f13-4c89-98c4-a13731a23f1d

AP Description & Expectations ๐ŸŽฏ

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

  • Essential Knowledge FUN-6.F.1: Some rational functions can be decomposed into sums of ratios of linear, non-repeating factors to which basic integration techniques can be applied.

Integrating Rational Functions Using Partial Fractions ๐Ÿ’ก

Let's consider the following integral:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(596).png?alt=media&token=014ae5e1-93b2-4892-9a93-f5207a8e5507

The integrand is a rational function with two linear factors in the denominator. With our current techniques, we canโ€™t solve this integral. However, by using partial fractions, we can split the rational integrand into two separate fractions that we know how to handle.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(599).png?alt=media&token=96c0d8d0-adb6-4acb-ae39-2373182ede7d

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(600).png?alt=media&token=10859276-d6f6-44fe-adf5-7c0ff7627bab

For the AP Calculus BC exam, you will only be dealing with examples where the denominator can be factored into non-repeating linear factors. A non-repeating linear factor is a term of the form (Ax+B) where A and B are numbers, and each term only shows up once in the denominator.

Steps to Using Partial Fraction Decomposition ๐Ÿ 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(602).png?alt=media&token=efdeb495-da28-4da2-ad84-ef1eb47d64be

Note: To find the values of A, B, Cโ€ฆ, you may need to use algebra techniques such as a system of equations, a matrix, or matching of coefficients.ย 

Note: Sometimes the numerator may not be just a number. It may be a linear or quadratic term. If the degree of the numerator is less than the denominator, you can use partial fractions immediately. If the numerator is the same or a higher degree, you will need to use polynomial long division or synthetic division until your numerator is one degree less than the denominator.

Worked Examples ๐Ÿ“”

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(605).png?alt=media&token=6e69c37d-6577-4967-8972-b2dc5cd1db7a

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(606).png?alt=media&token=c25db468-a07c-44ef-9674-9547d4a6f710

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(608).png?alt=media&token=10d51e47-eeda-49d3-a086-b4cf075255f6


https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(612).png?alt=media&token=7e125253-c081-457a-996d-aa8f670a8fd4

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(613).png?alt=media&token=c57ce087-56f6-4e70-be97-a86e1a000997

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(614).png?alt=media&token=ba078241-be6f-4309-8e17-6ec7c64742f9

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