Study materials›Calculus II›Partial Fractions
➗CALCULUS II STUDY GUIDE

Partial Fractions

Split rational functions by denominator type, solve coefficients, and find antiderivatives of the resulting terms.

8 key ideas8 flashcards5 practice questions
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EXAM SCOPEOpenStax Calculus Volume 2, partial fractions

This review follows the current course framework.

Checked against OpenStax Calculus Volume 2 ↗
DENOMINATOR SORT

Each factor gets the numerator form it needs.

x−a

constant numerator

(x−a)ᵐ

one term for every power

x²+bx+c

linear numerator if irreducible

SMART NOTES

Key ideas to know

Start with the relationships between ideas. Then close the notes and explain each one from memory.

  1. 01

    Polynomial division comes first when the numerator degree is at least the denominator degree.

  2. 02

    A proper rational function has numerator degree below denominator degree.

  3. 03

    Each distinct linear factor contributes a constant numerator.

  4. 04

    A repeated linear factor needs one term for every power up to its multiplicity.

  5. 05

    An irreducible quadratic factor needs a linear numerator.

  6. 06

    Coefficients can be found by substitution, coefficient matching, or both.

  7. 07

    Linear denominator terms produce logarithmic antiderivatives.

  8. 08

    Quadratic terms may require completing the square, substitution, or an inverse tangent form.

PART OF THE CALCULUS II GUIDE

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COMMON MIX-UPS

Two ideas worth correcting now

NOT QUITE

Every denominator factor receives a constant numerator

→
USE THIS INSTEAD

An irreducible quadratic receives a linear numerator.

NOT QUITE

Repeated factors need only the highest power

→
USE THIS INSTEAD

Every power through the multiplicity must appear.

ACTIVE RECALL

Flashcards

Answer before opening each card. The effort to retrieve is part of the learning.

1When is polynomial division required?Show answer ＋

When the numerator degree is at least the denominator degree.

2What term matches a distinct factor x−a?Show answer ＋

A divided by x−a.

3What terms match (x−a)²?Show answer ＋

A/(x−a) + B/(x−a)².

4What numerator matches x²+bx+c when it does not factor over the reals?Show answer ＋

A linear numerator Ax+B.

5What is an antiderivative of 1/(x−a)?Show answer ＋

ln|x−a| + C.

6Why multiply by the common denominator?Show answer ＋

It clears fractions and creates a polynomial identity.

7What can strategic substitution do?Show answer ＋

Set a factor to zero and isolate one coefficient.

8How is the decomposition checked?Show answer ＋

Combine the terms and compare with the starting rational function.

PRACTICE QUESTIONS

Explain it in your own words

Use the answer as a check after you have written or spoken your response.

01Decompose 1/((x−1)(x+2)).

A/(x−1)+B/(x+2), with A = 1/3 and B = −1/3.

02Why does a repeated factor need several terms?

Omitting lower powers leaves too few coefficients to represent every possible numerator.

03How does an irreducible quadratic differ from a linear factor?

Its numerator must have degree one below the quadratic, so Ax+B is used.

04What happens after decomposition?

Find each simpler antiderivative with logarithmic, substitution, or inverse tangent forms.

05Why should the final answer use absolute bars on linear logarithms?

The derivative of ln|x−a| equals 1/(x−a) on either side of a.

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