A-Level Maths: integration techniques you need to know
Integration is one of the most challenging topics in A-Level Maths, but it's also one of the most rewarding once it clicks. This article covers the key integration techniques you'll need, with worked examples for each method.
“Integration is the reverse of differentiation — but the techniques you need are far more varied. That's what makes it interesting.”
JJames Thornton
Maths Lead at Cognito
“Integration is the reverse of differentiation — but the techniques you need are far more varied. That's what makes it interesting.”
JJames Thornton
Maths Lead at Cognito
Integration by substitution
Substitution is often the first technique you learn. The idea is to simplify the integral by replacing a complicated expression with a single variable. Look for a function and its derivative within the integrand — that's your cue to use substitution.
For example, to integrate 2x(x² + 1)³, let u = x² + 1, so du = 2x dx. The integral becomes ∫u³ du, which is straightforward.
Always remember to change the limits of integration when using substitution in a definite integral. Converting back to the original variable at the end is an alternative, but changing limits is usually cleaner.
Integration by parts
Integration by parts is based on the product rule for differentiation. The formula is ∫u dv = uv − ∫v du. The trick is choosing which part of the integrand to call u and which to call dv.
A useful mnemonic is LIATE: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Choose u from whichever category comes first in this list.
| u (differentiate) | dv (integrate) | When to use |
|---|---|---|
| ln x | dx | Integrals involving ln x |
| x | eˣ dx | Products of polynomials and exponentials |
| x | sin x dx or cos x dx | Products of polynomials and trig functions |
| eˣ | sin x dx or cos x dx | Products of exponentials and trig (apply twice) |
Partial fractions
When you need to integrate a rational function (a fraction where both numerator and denominator are polynomials), partial fractions can break it into simpler pieces. First, factorise the denominator, then express the fraction as a sum of simpler fractions — each of which can be integrated directly.
If the degree of the numerator is greater than or equal to the degree of the denominator, you must perform polynomial long division first before splitting into partial fractions.
Choosing the right technique
One of the hardest parts of integration at A-Level is recognising which technique to use. Here's a decision framework.
If the integrand contains a composite function and the derivative of the inner function is also present (possibly with a constant multiple), use substitution. For example, ∫cos(3x) dx or ∫2xe^(x²) dx.
If you're integrating something like x·sin x or x²·eˣ, use integration by parts. Apply the LIATE rule to decide which factor to differentiate and which to integrate.
If the integrand is a fraction with polynomials top and bottom, try partial fractions. Factorise the denominator first, then decompose into simpler fractions.
Integrals like ∫sin²x dx or ∫cos²x dx require trig identities (double angle formulae) to rewrite them in an integrable form. Also look out for ∫tan²x dx = ∫(sec²x − 1) dx.
Roughly
25%
of A-Level Maths Paper 1 marks typically involve integration
Practice problems
The only way to get confident with integration is to practise. Work through these problems, then check your answers against the mark scheme.
Pattern recognition
The more integrals you solve, the faster you'll recognise which technique to apply.
Speed under pressure
In the exam, you can't afford to spend 10 minutes deciding on a method. Practice builds automaticity.
Error spotting
Common mistakes — forgetting the constant, sign errors, wrong limits — become obvious when you've seen them before.
