Integration by recognition

The ln pattern

∫ f'(x)/f(x) dx = ln|f(x)| + c.

If the top is a multiple of the derivative of the bottom, adjust the constant: ∫ x/(x² + 5) dx = (1/2)ln(x² + 5) + c.

tan x = sin x/cos x, so ∫ tan x dx = -ln|cos x| + c = ln|sec x| + c.

The power pattern

∫ f'(x)(f(x))ⁿ dx = (f(x))^{n + 1}/(n + 1) + c, for n not equal to -1.

For example ∫ x(x² + 1)⁴ dx: try (x² + 1)⁵, which differentiates to 10x(x² + 1)⁴, so the answer is (1/10)(x² + 1)⁵ + c.

Try, then adjust

Guess the answer, differentiate it, and compare with what you want. Fix the constant by multiplying. Also: ∫ f'(x)e^(f(x)) dx = e^(f(x)) + c.

Worked example

Find ∫ x/(x² + 5) dx.

1. d/dx (x² + 5) = 2x, and the top is x = (1/2)(2x).
2. So the integral is (1/2)ln|x² + 5|.
3. x² + 5 is always positive, so the modulus can be dropped.

Answer: (1/2)ln(x² + 5) + c

Check yourself

1. Find ∫ 2x/(x² + 3) dx.
   • 2ln(x² + 3) + c
   • x²/(x³/3 + 3x) + c
   • ln(x² + 3) + c
   • 1/(x² + 3) + c
   Show answer

   ln(x² + 3) + c. The top is exactly the derivative of the bottom.

2. Find ∫ tan x dx.
   • ln|sec x| + c
   • sec² x + c
   • ln|sin x| + c
   • -ln|sec x| + c
   Show answer

   ln|sec x| + c. tan x = sin x/cos x = -(-sin x)/cos x, giving -ln|cos x| = ln|sec x|.

3. Find ∫ x(x² + 1)⁴ dx.
   • (1/5)(x² + 1)⁵ + c
   • (x²/2)(x² + 1)⁵/5 + c
   • (1/10)(x² + 1)⁵ + c
   • 2(x² + 1)⁵ + c
   Show answer

   (1/10)(x² + 1)⁵ + c. d/dx (x² + 1)⁵ = 10x(x² + 1)⁴, so divide by 10.

4. Work out the exact value of ∫₀^{1} x/(x² + 1) dx and give it to 4 decimal places.
   Show answer

   0.3466. The integral is (1/2)ln(x² + 1). (1/2)ln 2 - 0 = 0.3466.

5. Evaluate ∫₀^{π/2} sin x cos³ x dx. Give your answer as a fraction in its simplest form.
   Show answer

   1/4. d/dx (cos⁴ x) = -4cos³ x sin x, so the integral is -(1/4)cos⁴ x. From 0 to π/2: 0 - (-1/4) = 1/4.

6. Find ∫ cos x/(1 + sin x) dx.
   • ln|cos x| + c
   • ln|1 + sin x| + c
   • sin x/(x + cos x) + c
   • -ln|1 + sin x| + c
   Show answer

   ln|1 + sin x| + c. d/dx (1 + sin x) = cos x, which is the top.

7. By spotting a derivative, find the value of ∫₀^{1} 2x e^x2 dx to 3 decimal places.
   Show answer

   1.718. 2x is the derivative of x², so the integral is e^x2. e¹ - e⁰ = 1.718.

8. Find ∫ sec² 2x dx.
   • 2tan 2x + c
   • tan 2x + c
   • (1/2)tan 2x + c
   • (1/3)sec³ 2x + c
   Show answer

   (1/2)tan 2x + c. d/dx (tan 2x) = 2sec² 2x, so halve it.

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Sources

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

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