A level Maths integration

By The STEM-Tutor team · Updated October 2026

Integration reverses differentiation and finds areas. Here are the techniques A level Maths expects and how to choose between them.

Standard integrals

• ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c, for n ≠ −1
• ∫1/x dx = ln|x| + c
• ∫e^kx dx = (1/k)e^kx + c
• ∫sin kx dx = −(1/k)cos kx + c and ∫cos kx dx = (1/k)sin kx + c

Never forget + c on an indefinite integral.

Definite integrals and area

A definite integral gives the signed area between a curve and the x-axis. Areas below the axis come out negative, so split the integral where the curve crosses the axis when you need total area.

Techniques

• Substitution: replace an inner function with u. Change the limits too for definite integrals.
• Integration by parts: ∫u(dv/dx) dx = uv − ∫v(du/dx) dx. Choose u as the part that gets simpler when differentiated.
• Partial fractions: split a rational function into simpler fractions first.
• Using identities: for example, rewrite sin²x using a double-angle formula.

Numerical integration

The trapezium rule estimates an area using strips. It overestimates when the curve is convex and underestimates when it is concave.

Differential equations

First-order differential equations with separable variables are solved by separating and integrating both sides. Use given conditions to find the constant.

Choosing a method

1. Is it a standard integral? Integrate directly.
2. Is there a function and its derivative? Try substitution.
3. Is it a product of different types, such as x e^x? Try by parts.
4. Is it a fraction with a factorised denominator? Try partial fractions.

Integration as reverse differentiation and area

Integration has two meanings that turn out to be the same idea. As the reverse of differentiation, it finds a function from its gradient. As an area, it adds up thin strips under a curve. The Fundamental Theorem of Calculus connects the two, which is why you can find areas by evaluating an antiderivative at two limits.

Worked examples

Example 1: polynomials

Find ∫(6x² + 4x) dx.

• 2x³ + 2x² + c.

Example 2: a definite integral

Evaluate ∫₁³ x² dx.

• [x³/3] from 1 to 3 = 9 − 1/3 = 26/3.

Example 3: area between a curve and the axis

Find the area enclosed by y = 4 − x² and the x-axis.

• Roots at x = ±2. Area = ∫₋₂² (4 − x²) dx = [4x − x³/3] from −2 to 2 = 32/3.

Example 4: substitution

Find ∫2x(x² + 1)³ dx.

• Let u = x² + 1, so du = 2x dx. ∫u³ du = u⁴/4.
• Answer: (x² + 1)⁴/4 + c.

Example 5: integration by parts

Find ∫x e^x dx and ∫ln x dx.

• u = x, dv/dx = e^x: x e^x − e^x + c.
• Write ln x as 1 × ln x with u = ln x: x ln x − x + c.

Example 6: partial fractions

Find ∫1/((x + 1)(x + 2)) dx.

• 1/((x + 1)(x + 2)) = 1/(x + 1) − 1/(x + 2).
• Answer: ln|x + 1| − ln|x + 2| + c.

Example 7: using an identity

Find ∫sin²x dx.

• Use sin²x = ½(1 − cos 2x).
• Answer: x/2 − (sin 2x)/4 + c.

Example 8: a separable differential equation

Solve dy/dx = xy with y = 2 when x = 0.

• Separate: (1/y)dy = x dx, so ln|y| = x²/2 + c.
• y = Ae^x²/2; y(0) = 2 gives A = 2, so y = 2e^x²/2.

Common mistakes

• Forgetting + c on indefinite integrals.
• Not changing the limits after a substitution.
• Adding signed areas when the curve crosses the axis, instead of splitting the integral.
• Choosing u badly in integration by parts; pick the part that simplifies when differentiated.

Practice questions

Try each one before opening the answer.

Frequently asked questions

How do I choose an integration method?

Check standard results first, then look for a function and its derivative (substitution), a product of different types (by parts), or a rational function (partial fractions).

Is the trapezium rule part of integration?

Yes. It gives a numerical estimate when exact integration is hard, and you should know when it over- or underestimates.

Key takeaways

• Integration reverses differentiation and finds areas.
• Always add + c to indefinite integrals.
• Learn substitution, by parts, partial fractions and identities.
• Split areas where the curve crosses the axis.
• Separate variables to solve first-order differential equations.

Where to go next

Integration links closely to differentiation and to motion in mechanics.

• A level Maths differentiation
• A level Maths mechanics
• A level Maths formula booklet

Understanding area and signed area

A definite integral gives signed area. Area above the x-axis counts as positive and area below counts as negative. If a curve crosses the axis between your limits, the positive and negative parts can cancel, and a single integral may give a much smaller answer than the true area, or even zero. To find the total area, find where the curve crosses the axis, integrate each part separately, and add the sizes of the results.

The same idea matters in mechanics. Integrating velocity gives displacement, which can be negative when the particle moves backwards. Distance travelled is different: it is the total of the sizes of each part of the motion.

A revision routine for integration

Integration rewards pattern recognition more than any other A level topic. Keep a list of the forms you have met, such as f'(x)/f(x), a function times its inner derivative, or a polynomial times an exponential, together with the method that suits each. Practise short drills of ten integrals at a time and check every answer by differentiating it; if you get back the original, you are right. Then move on to exam-style questions where integration is combined with areas, differential equations or the trapezium rule.

The trapezium rule in practice

The trapezium rule splits the area into strips of equal width h and treats the top of each strip as a straight line. The estimate is h/2 times the first and last heights plus twice the sum of the heights in between. If the curve is concave up, the straight tops lie above the curve, so the rule overestimates. If the curve is concave down, it underestimates. Using more, thinner strips makes the estimate more accurate, and exam questions often ask you to explain this.

Integration takes time to master, so be patient with yourself and keep practising little and often.