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๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

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

6.12 Using Linear Partial Fractions

#partialfractions

#integration

#bconly

โฑ๏ธย ย 2 min read

written by

Catherine Liu

catherine liu


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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:

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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.

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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 

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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“”

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