CalculusLesson 3 of 5 in the course Techniques of integration

Partial Fractions: Factor Templates, Coefficients, and Integration

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

Partial fraction decomposition rewrites a rational function as a polynomial plus simpler rational terms. This makes complicated expressions easier to analyze and integrate.

A rational function is a quotient of polynomials, such as f(x)g(x)\frac{f(x)}{g(x)}. A partial fraction decomposition expresses it in a form such as

f(x)g(x)=p(x)+∑jfj(x)gj(x).\frac{f(x)}{g(x)}=p(x)+\sum_j\frac{f_j(x)}{g_j(x)}.

All flashcards

What is a rational function?
A quotient of two polynomials, with a nonzero denominator polynomial.
Define a proper rational fraction.
It is a rational fraction whose numerator degree is smaller than its denominator degree.
What does polynomial long division provide?
It writes the numerator as quotient times denominator plus a remainder of smaller degree than the denominator.
What is the polynomial part in a decomposition?
The polynomial quotient produced by division; it may be zero when the original fraction is proper.
Why use partial fractions?
They turn a rational function into simpler terms that support computations such as integration, series expansion, and transforms.

Quiz questions

  1. For a proper fraction with distinct linear denominator factors (x−a)(x−b)(x-a)(x-b), which partial-fraction template is appropriate?Answer: Ax−a+Bx−b\frac{A}{x-a}+\frac{B}{x-b}
  2. If 1(x+3)(x−1)=Ax+3+Bx−1\frac{1}{(x+3)(x-1)}=\frac{A}{x+3}+\frac{B}{x-1}, what identity results from clearing denominators?Answer: 1=A(x−1)+B(x+3)1=A(x-1)+B(x+3)
  3. For a proper rational function P(x)/Q(x)P(x)/Q(x) with a simple root α\alpha of QQ, what is the coefficient of 1/(x−α)1/(x-\alpha)?Answer: P(α)Q′(α)\frac{P(\alpha)}{Q'(\alpha)}
  4. Why is evaluating at a repeated pole insufficient to determine every partial-fraction coefficient?Answer: The decomposition contains coefficients for several powers of the repeated factor.
  5. Which standard antiderivative forms can arise from partial fractions over an irreducible real quadratic?Answer: Logarithmic and arctangent forms

Based on the Wikipedia article Partial fraction decomposition, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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