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Unit 6

6.12 Using Linear Partial Fractions

2 min readβ€’april 26, 2020

catherine

Catherine Liu


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AP Description & Expectations 🎯

  • 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 πŸ’‘

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 can’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 🏠

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, C…, you may need to use algebra techniques such as a system of equations, a matrix, or matching of coefficients.Β 
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 πŸ“”

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

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