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 💡
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 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.
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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 🏠
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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.
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 📔
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