Integration for A-level Maths
Integration in A-level Maths is the reverse of differentiation, and it finds areas. You need the standard integrals (xⁿ, eᵏˣ, 1/x, sin kx and cos kx), definite integrals for areas under and between curves, and three methods for harder integrals: substitution, integration by parts and partial fractions. It finishes with solving separable differential equations.
What A-level Maths integration covers
AQA's A-level Maths specification lists eight integration statements, H1 to H8. In short:
| Statement | What you need to do |
|---|---|
| H1 | Know and use the Fundamental Theorem of Calculus |
| H2 | Integrate xⁿ (n ≠ −1), eᵏˣ, 1/x, sin kx and cos kx, and their sums and multiples |
| H3 | Evaluate definite integrals; find the area under a curve and between two curves |
| H4 | Understand integration as the limit of a sum |
| H5 | Integrate by substitution and by parts, as the reverse of the chain and product rules |
| H6 | Integrate using partial fractions with linear denominators |
| H7 | Solve simple first order separable differential equations, including particular solutions |
| H8 | Interpret the solution of a differential equation in context |
If integration is new to you, what is an integral explains the idea from the start.
Standard integrals
| Function | Integral (add + C) |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ ÷ (n + 1) |
| eᵏˣ | (1/k) eᵏˣ |
| 1/x | ln |x| |
| sin kx | −(1/k) cos kx |
| cos kx | (1/k) sin kx |
Every indefinite integral needs + C, because differentiating a constant gives zero, so any constant could have been there. The trig results only work with x in radians.
Example 1: polynomials
Raise each power by one and divide by the new power: 6x² becomes 6x³/3 = 2x³.
Example 2: the other standard integrals
Definite integrals and areas
A definite integral has limits. Integrate, then subtract the value at the lower limit from the value at the upper limit; no + C is needed. For example:
The area between a curve and the x-axis is the definite integral, as long as the curve is above the axis. Area below the axis comes out negative, so split the integral where the curve crosses.
Example 3: area between two curves
Find the area enclosed by the curve y = 4x − x² and the line y = x.
- Where they meet: 4x − x² = x gives x² − 3x = 0, so x = 0 or x = 3.
- Between those, the curve is above the line, so integrate (top − bottom): (4x − x²) − x = 3x − x².
- Area = [1.5x² − x³/3] from 0 to 3 = (13.5 − 9) − 0 = 4.5 square units.
Try one: evaluate a definite integral, then press Mark to check it in the Calculus course.
Evaluate a definite integral, giving a numeric or fraction answer.
Press Mark to have your answer marked in the Calculus course.
Integration methods
Integration by substitution
Use it when the integral contains a function and (a multiple of) its derivative. For ∫ 2x(x² + 1)⁴ dx, let u = x² + 1, so du = 2x dx:
Integration by parts
Use it for a product of two different kinds of function, using ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. For ∫ x cos x dx, take u = x and dv/dx = cos x, so v = sin x:
The integration by parts guide covers how to choose u and when to use the method twice.
Partial fractions
Split a fraction with a factorised denominator into simpler fractions, then integrate each as a log. Since 5/((x − 1)(x + 4)) = 1/(x − 1) − 1/(x + 4):
Separable differential equations
Solve dy/dx = 3x²y, given y = 2 when x = 0. Separate the variables: ∫ (1/y) dy = ∫ 3x² dx, so ln |y| = x³ + c and y = Ae^(x³). Putting in y = 2 at x = 0 gives A = 2, so y = 2e^(x³).
Which method should I use?
| If the integral looks like | Try |
|---|---|
| A sum of powers, eᵏˣ, 1/x, sin kx or cos kx | Standard integrals |
| f(g(x)) × g′(x), a function times its inner derivative | Substitution (or reverse chain rule) |
| A product of different types, like x eˣ or x sin x | Integration by parts |
| A fraction with a factorised denominator | Partial fractions |
| dy/dx = f(x)g(y) | Separate the variables |
| A definite integral you can't do exactly | The trapezium rule |
For integrals that can't be done exactly, the trapezium rule gives an estimate. The integral calculator shows the steps for any of these methods.
Common mistakes
- Forgetting + C on an indefinite integral.
- Dividing by the old power. ∫ x³ dx = x⁴/4, not x⁴/3.
- Getting the sign wrong on sin. ∫ sin x dx = −cos x + C.
- Using degrees. Trig integrals need radians.
- Losing area below the axis. Split the integral where the curve crosses the x-axis.
- Leaving x in after substituting. Change everything, including dx and the limits, into u.
To practise integration in a short timed paper, try today's A-level Maths: Calculus paper.
Common questions
What integration is in A-level Maths?
What is the integral of 1/x?
What is the integral of e^(kx)?
How do you find the area between two curves?
When do you use integration by parts?
Why do you add + C?
Revision cards
Say the answer first, then flip the card to check it.
Where to go next
- Solve it
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Maths · CalculusIntegration by parts: formula, LIATE and worked examplesThe integration by parts formula, how to choose u with LIATE, the tabular method, and worked examples from x e^x to e^x sin x, plus definite integrals.