Integrating with partial fractions

Linear factors

∫ A/(ax + b) dx = (A/a)ln|ax + b| + c.

So 2/(x - 1) + 3/(x + 2) integrates to 2ln|x - 1| + 3ln|x + 2| + c.

Repeated and improper

• A repeated factor gives a power, not a log: ∫ 1/(x + 3)² dx = -1/(x + 3) + c.
• For an improper fraction, divide or include the polynomial part first, then integrate each term.
• x/(x² + 5) is not split: it is a recognition integral, (1/2)ln(x² + 5).

Tidy answers

Combine logs with the log laws: 2ln 2 + 3ln(5/4) = ln(4 × 125/64) = ln(125/16). Never integrate the top and bottom of a fraction separately.

Worked example

Evaluate ∫₂^{3} (5x + 1)/((x - 1)(x + 2)) dx.

1. Partial fractions: 2/(x - 1) + 3/(x + 2)
2. Integrate: 2ln|x - 1| + 3ln|x + 2|
3. (2ln 2 + 3ln 5) - (0 + 3ln 4)

Answer: 2ln 2 + 3ln(5/4) = ln(125/16) ≈ 2.056

Check yourself

1. Find ∫ (3/(x - 1) - 2/(x + 2)) dx.
   • 3ln|x - 1| + 2ln|x + 2| + c
   • 3ln|x - 1| - 2ln|x + 2| + c
   • -3/(x - 1)² + 2/(x + 2)² + c
   • ln|3x - 3| - ln|2x + 4| + c
   Show answer

   3ln|x - 1| - 2ln|x + 2| + c. Each term integrates to a constant times ln.

2. Evaluate ∫₃^{5} (5x - 4)/((x + 1)(x - 2)) dx, to 3 decimal places.
   Show answer

   3.414. (5x - 4)/((x + 1)(x - 2)) = 3/(x + 1) + 2/(x - 2). The integral is 3ln|x + 1| + 2ln|x - 2|. From 3 to 5: 3ln(6/4) + 2ln 3 = 3.414.

3. Find ∫ 1/(x + 3)² dx.
   • ln|x + 3|² + c
   • -1/(x + 3) + c
   • 1/(x + 3) + c
   • 2ln|x + 3| + c
   Show answer

   -1/(x + 3) + c. (x + 3)^-2 integrates to -(x + 3)^-1.

4. Split 1/((x + 1)(x + 2)) into partial fractions and hence find ∫₀^{1} 1/((x + 1)(x + 2)) dx to 3 decimal places.
   Show answer

   0.288. 1/((x + 1)(x + 2)) = 1/(x + 1) - 1/(x + 2). The integral is ln|x + 1| - ln|x + 2|. From 0 to 1: (ln 2 - ln 3) - (0 - ln 2) = ln(4/3) = 0.288.

5. Which is an integral of 2/(2x + 1)?
   • ln|2x + 1| + c
   • 2ln|2x + 1| + c
   • (1/2)ln|2x + 1| + c
   • 4ln|2x + 1| + c
   Show answer

   ln|2x + 1| + c. ∫ A/(ax + b) dx = (A/a)ln|ax + b| with A = 2, a = 2.

6. This fraction is not split into partial fractions. Find ∫ x/(x² + 5) dx.
   • ln(x² + 5) + c
   • x²/(2(x³/3 + 5x)) + c
   • (1/2)ln(x² + 5) + c
   • 2ln(x² + 5) + c
   Show answer

   (1/2)ln(x² + 5) + c. The top is half the derivative of the bottom, 2x.

7. The improper fraction (x² + 3x + 4)/((x + 1)(x + 2)) equals 1 + 2/(x + 1) - 2/(x + 2). Find its integral from x = 0 to x = 1, to 3 decimal places.
   Show answer

   1.575. The fraction is 1 + 2/(x + 1) - 2/(x + 2). The integral is x + 2ln|x + 1| - 2ln|x + 2|. From 0 to 1: 1 + 2ln 2 - 2ln 3 + 2ln 2 = 1.575.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.