Integration A-Level Maths: The Complete Guide (With Examples)

Why integration is the defining topic of A-Level Maths

No other topic appears as consistently, carries as many marks, or separates grades as definitively as integration. A student who truly understands integration — not just procedure, but reasoning — is almost always working at A or A* level.

Integration is also the topic most likely to reward confident technique. The method marks are often independent — meaning if you choose the right technique and execute the first step correctly, you pick up marks even if you make an error later.

The integration decision tree

Before you integrate anything, ask these questions in order:

1. Is it a standard function?

If you can immediately recognise it — eˣ, sin x, cos x, 1/x, xⁿ — integrate directly. Don't overthink it.

2. Is there a composite function where the derivative of the inner function appears?

If yes: reverse chain rule (also called integration by recognition or substitution by observation).

Example: ∫2x·eˣ² dx — notice that 2x is the derivative of x², so the answer is eˣ² + c.

3. Is it a product of two unrelated functions?

If yes: integration by parts.

Formula: ∫u dv = uv − ∫v du

Choose u using LIATE order (Logarithm, Inverse trig, Algebraic, Trig, Exponential) — pick the function that simplifies when differentiated.

4. Does it involve a sum of partial fractions?

If the integrand is a rational function (polynomial over polynomial), use partial fractions first, then integrate each term.

Example: ∫(2x+1)/((x+1)(x−2)) dx → split into A/(x+1) + B/(x−2) first.

5. Does it involve a trigonometric expression that can be simplified?

Use double angle formulae to convert sin²x or cos²x into integrable form.

Example: ∫sin²x dx → ∫(1 − cos2x)/2 dx

6. Is substitution needed?

For everything else. Substitution works when you can choose a u that simplifies the integrand significantly. For definite integrals, change the limits too.

Integration by parts: the most common errors

Error 1: Wrong choice of u

Choosing u = eˣ when u = x is correct leads to an integral that gets more complex, not less. If your IBP is making the integral worse, swap u and dv.

Error 2: Sign errors on the second integral

∫u dv = uv − ∫v du — the minus sign applies to the entire second integral, including any signs within it.

Error 3: Forgetting +c on indefinite integrals

This is a B mark in most mark schemes — dropping it costs you a mark for what is genuinely a correct answer otherwise.

Integration by substitution: the most common errors

Error 1: Not changing limits on definite integrals

If you substitute u = x² + 1 and your x-limits are 0 and 2, your u-limits become 1 and 5. Students who forget this get the method marks but lose accuracy marks.

Error 2: Not cancelling dx correctly

If u = x² + 1, then du/dx = 2x, so dx = du/2x. The 2x must cancel with a 2x in the numerator — if it doesn't, your substitution is wrong.

Definite integration and area

Area between a curve and the x-axis:

• If the curve is entirely above the x-axis: A = ∫[a to b] y dx
• If the curve dips below: split at the x-axis crossing and add the absolute values

Area between two curves:

A = ∫[a to b] (f(x) − g(x)) dx where f(x) is the upper curve.

The most common error here: students forget to subtract, or subtract in the wrong order, and get a negative area. Area is always positive — if your answer is negative, you've subtracted the wrong way around.

How to practise integration effectively

The only way to build integration fluency is through volume of practice on unseen questions. Reading examples doesn't work — you need to be placed in front of an integral and required to identify the technique without prompting.

Generate integration questions on Infinity Stars at escalating difficulty: start with standard forms, move to by-parts and substitution, then mixed questions where the technique isn't labelled. The method guide after each question shows you the exact decision process — not just the answer.

Integration on your paper is not a question of whether you've seen the type before. It's whether you can execute the decision tree under pressure.