A-Level Maths: Integration

Integration at A-Level is a technique-selection game: the mark scheme rewards choosing the right method before executing it cleanly. This quiz covers the full method toolkit and the judgement about when each applies.

The technique questions centre on the specification's two big methods. Integration by parts appears twice in its canonical exam forms: ∫x eˣ dx with the u = x choice, and the definite classic ∫₁ᵉ ln x dx — the integral that looks impossible until parts turns it into [x ln x − x], evaluating to exactly 1. Substitution gets a method-selection question on ∫x√(x+1) dx, where both u = x + 1 and u² = x + 1 clear the root while superficially similar choices do nothing.

A parallel judgement question covers ∫sin²x dx: the power-reduction identity from the cosine double angle formula is the route, while rewriting as 1 − cos²x — plausible-looking — leads nowhere. The structural questions test the definite integral's actual properties (limit reversal changes sign, additivity over adjacent intervals) against two carefully false claims, and the constant of integration's real meaning: information lost to differentiation, recovered through boundary conditions.

Area interpretation appears through the region bounded by y = x² and y = 4, with the symmetric-halves formulation tested alongside the direct one. Differential equations close the set: separating dy/dx = xy, integrating both sides and applying y(0) = 1 to reach y = e^(x²/2) — the complete separable workflow in one question.

Every key has been checked.

Choose and execute integration by parts, including the ln x classic
Select substitutions that genuinely simplify a given integrand
Use power-reduction identities for squared trig integrals
Apply the definite integral's properties: limit reversal and additivity
Solve separable differential equations with initial conditions

Topic scope follows section H (Integration) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: integration as the reverse of differentiation, definite integrals and areas, substitution, integration by parts, and separable differential equations.

Sample question

The parabola y = x squared and a dashed horizontal line (y = 4) meeting it at two points symmetric about the y-axis.

Which of the following integrals correctly represent the area bounded by $y = x^2$ and $y = 4$?

$\int_{-2}^{2} (4 - x^2) dx$
$\int_{0}^{4} (4 - x^2) dx$
$2 \int_{0}^{2} (4 - x^2) dx$
$\int_{-2}^{2} (x^2 - 4) dx$

See the answer

$\int_{-2}^{2} (4 - x^2) dx$, $2 \int_{0}^{2} (4 - x^2) dx$

The curves intersect at $x = \pm 2$. The area is $\int_{-2}^{2} (4 - x^2) dx$, which is also equal to $2 \int_{0}^{2} (4 - x^2) dx$ due to symmetry.

Try this quiz →Practice these flashcards →Try this written work →

← Mathematics

↑ A-Level

A levels are awarded by AQA, Pearson Edexcel, OCR, WJEC/Eduqas and CCEA, and regulated by Ofqual, Qualifications Wales and CCEA Regulation, the regulators for England, Wales and Northern Ireland. Zestly is not affiliated with or endorsed by Ofqual, UCAS or any awarding body. These materials are our own and reproduce no past-paper content.