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11.7PureStretch

Integration Using Partial Fractions

A rational function that resists direct integration can be split into partial fractions, each of which integrates to a logarithm. This connects algebra and calculus neatly.

26 min Integration
Edexcel A level Maths: 11.7 Integrating Using Partial FractionsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Split a rational function into partial fractions
  • Integrate each term to a logarithm
  • Combine logs with the laws of logarithms
  • Evaluate definite integrals
1

The method

Decompose into partial fractions, then integrate each Ax+a\dfrac{A}{x + a} term to Aln⁡∣x+a∣A\ln|x + a|. Combine the logarithms at the end.

∫Ax+a dx=Aln⁡∣x+a∣+c\int \frac{A}{x+a}\,dx = A\ln|x+a| + c
Each partial fraction integrates to a log.
Example 1Find ∫1x(x+1) dx\int \dfrac{1}{x(x+1)}\,dx.
1Partial fractions: 1x−1x+1\tfrac{1}{x} - \tfrac{1}{x+1}.
2ln⁡∣x∣−ln⁡∣x+1∣+c=ln⁡∣xx+1∣+c\ln|x| - \ln|x+1| + c = \ln\left|\tfrac{x}{x+1}\right| + c.
Answer
ln⁡∣xx+1∣+c\ln\left|\dfrac{x}{x+1}\right| + c

Tip — Combine the logs with log laws to give a tidy single-log answer.

Formula recap

∫Ax+a dx=Aln⁡∣x+a∣+c\int \frac{A}{x+a}\,dx = A\ln|x+a| + c
Standard log integral.
ln⁡p−ln⁡q=ln⁡pq\ln p - \ln q = \ln\tfrac{p}{q}
Combine logs.

Common mistakes to avoid

Integrating the rational function directly.
Split into partial fractions first.
Dropping the modulus in the logarithm.
It is ln|x + a|.

Key takeaways

  • Decompose into partial fractions.
  • Each A/(x+a) integrates to A ln|x+a|.
  • Combine logs into a single tidy logarithm.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ∫Ax+a dx\int \frac{A}{x+a}\,dx equals:

    • Aln⁡∣x+a∣+cA\ln|x+a| + c
    • A(x+a)2+c\frac{A}{(x+a)^2}+c
    • A(x+a)+cA(x+a)+c
    • ln⁡(x+a)\ln(x+a)
    Show answer and explanation

    Answer: Aln⁡∣x+a∣+cA\ln|x+a| + c

    Standard log.

  2. 1x(x+1)\frac{1}{x(x+1)} in partial fractions is:

    • 1x−1x+1\frac{1}{x} - \frac{1}{x+1}
    • 1x+1x+1\frac{1}{x} + \frac{1}{x+1}
    • 1x+1−1x\frac{1}{x+1} - \frac{1}{x}
    • 2x\frac{2}{x}
    Show answer and explanation

    Answer: 1x−1x+1\frac{1}{x} - \frac{1}{x+1}

    Decompose.

  3. ∫1x(x+1) dx\int \frac{1}{x(x+1)}\,dx equals:

    • ln⁡∣xx+1∣+c\ln\left|\frac{x}{x+1}\right| + c
    • ln⁡∣x(x+1)∣+c\ln|x(x+1)| + c
    • 1x+c\frac{1}{x}+c
    • ln⁡∣x+1∣+c\ln|x+1| + c
    Show answer and explanation

    Answer: ln⁡∣xx+1∣+c\ln\left|\frac{x}{x+1}\right| + c

    Combine the two logs.

  4. Before integrating a rational function, you should:

    • split into partial fractions
    • differentiate
    • square it
    • use radians
    Show answer and explanation

    Answer: split into partial fractions

    Each piece is a log.

  5. ln⁡∣x∣−ln⁡∣x+1∣\ln|x| - \ln|x+1| combines to:

    • ln⁡∣xx+1∣\ln\left|\frac{x}{x+1}\right|
    • ln⁡∣x(x+1)∣\ln|x(x+1)|
    • ln⁡∣x∣⋅ln⁡∣x+1∣\ln|x| \cdot \ln|x+1|
    • ln⁡∣x+1∣\ln|x+1|
    Show answer and explanation

    Answer: ln⁡∣xx+1∣\ln\left|\frac{x}{x+1}\right|

    Subtraction → division.

Next step

Check your understanding

Work through questions on Integration Using Partial Fractions. Use the feedback to decide what needs another look.

1 past-paper question · 7 marks

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