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 — the question was sharpened during verification so only genuinely workable substitutions are keyed.

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.

Five items in this set were repaired during verification — duplicate options replaced and two keys extended where a "distractor" was mathematically valid — and every final key has been recomputed by hand.

• 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.