A-Level Integration Revision
Integration revision for A-Level and International A-Level Maths
Integration is the second half of A-Level calculus, and the half most students find harder — not because the individual techniques are difficult, but because choosing the right one is. A question that looks like it needs integration by parts often yields to the reverse chain rule in a single line, and recognising which is which only comes from volume. This free revision engine generates a new integration question every time and marks your answer instantly, so you can build that recognition rather than working through the same fifteen textbook questions twice. It covers the full integration content of Edexcel A-Level Maths and the International A-Level Pure 1 to Pure 4 papers.
What the integration revision engine covers
Fifteen topics, from the first introduction to the Year 13 calculus techniques:
Integration of xⁿ — the power rule for integration and the constant of integration Indefinite integrals — integrating sums, differences and rewritten roots and reciprocals Definite integrals — evaluating between limits and interpreting the result Area under a curve — including regions below the x-axis Area between two curves — finding the points of intersection first The trapezium rule — estimating an area and deciding whether it over- or under-estimates Integration of standard functions — eˣ, 1/x, sin x and cos x Integration of f(ax + b) — the linear substitution shortcut The reverse chain rule — spotting when a function and its derivative both appear Integration by substitution — choosing u and changing the limits Integration by parts — choosing u and dv/dx, and when to apply it twice Integration using partial fractions — splitting a rational function first Integration of trigonometric functions — using identities to make an integral doable Volumes of revolution — rotating a region about the x-axis or y-axis Differential equations — separating the variables and finding the particular solution
Integration formulas you need to know
The power rule. ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c, valid for every n except n = −1. The exception. ∫(1/x) dx = ln|x| + c. The modulus signs matter. Standard results. ∫eˣ dx = eˣ + c, ∫sin x dx = −cos x + c, ∫cos x dx = sin x + c. Linear inside. ∫f(ax + b) dx divides by a as well as integrating, so ∫e³ˣ dx = (1/3)e³ˣ + c. Definite integrals. Evaluate at the upper limit, subtract the value at the lower limit. The constant of integration cancels, so you can drop it. Area between two curves. Integrate (upper curve − lower curve) between their points of intersection. Integration by parts. ∫u(dv/dx) dx = uv − ∫v(du/dx) dx. Choose u to be the part that gets simpler when differentiated. Volumes of revolution. About the x-axis, V = π∫y² dx. About the y-axis, V = π∫x² dy.
How to use it
Pick a single topic or a whole exam module. Type your answer and the engine marks it instantly, comparing your expression with the correct integral mathematically — so an answer written with a different but equivalent arrangement of terms still counts as correct. Two hints are available per question: the first names the technique, the second walks you through the method without giving the answer away. Sign in and your accuracy is saved to your A-Level skill tree.
More A-Level Maths revision
Integration and differentiation are two halves of the same topic, and many integration questions are differentiation run backwards — if you are not fluent with the derivative of a function, integrating it is much harder. The differentiation revision engine is the natural companion. You can track your progress across every topic on the revision hub, and all our revision engines are free to use.