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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 Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.7 Integrating Using Partial FractionsWatch the full walkthrough 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

The method

Decompose into partial fractions, then integrate each $\frac{A}{x + a}$ term to $A ln ∣ x + a ∣$ . Combine the logarithms at the end.

\int \frac{A}{x + a} d x = A ln ∣ x + a ∣ + c

Each partial fraction integrates to a log.

1Partial fractions:

\frac{1}{x} - \frac{1}{x + 1}

ln ∣ x ∣ - ln ∣ x + 1∣ + c = ln \frac{x}{x + 1} + c

Answer

ln \frac{x}{x + 1} + c

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

Formula recap

\int \frac{A}{x + a} d x = A ln ∣ x + a ∣ + c

Standard log integral.

ln p - ln q = ln \frac{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

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