H: Integration

Overview

Integration turns a rate of change back into a quantity. In AQA A-level Mathematics 7357, H: Integration is not a single routine: it connects antidifferentiation, definite integrals, areas, the limit of a sum, substitution, parts, partial fractions and separable differential equations. The unifying idea is reversal. If differentiation tells you how quickly a quantity is changing, integration rebuilds the original function or total accumulated change.

This matters because Integration can be assessed in the Pure core on all three A-level papers. Each paper lasts 2 hours and carries 100 marks, so candidates should budget roughly 1.2 minutes per mark across the paper. Questions range from a one-mark antiderivative to a multi-stage modelling problem in which an integral, an initial condition and interpretation are all required. A strong answer is not merely an answer: it shows the method clearly enough for method credit to be awarded if arithmetic later slips. 1 2

Exam map: the AQA H1–H8 sequence

AQA requires the Fundamental Theorem of Calculus (H1), standard antiderivatives (H2), definite integrals and areas (H3), integration as a limit of a sum (H4), substitution and parts (H5), partial fractions with linear denominators (H6), separable first-order differential equations (H7), and contextual interpretation including kinematics (H8). The question to ask before writing anything is: what has differentiation produced, and which reverse process undoes it?

Key concepts

1. Fundamental Theorem of Calculus: the endpoint engine

If (F'(x)=f(x)), then

[\int_a^b f(x),dx=F(b)-F(a)=\left[F(x)\right]_a^b.]

The theorem says that a definite integral can be found from an antiderivative. The square bracket notation means “substitute the upper limit, then subtract the value at the lower limit”. It is not optional decoration: it protects the order of evaluation. For a definite integral, there is no arbitrary constant because any constant would cancel in (F(b)-F(a)).

The integral is signed area. A region above the (x)-axis contributes positively; one below contributes negatively. Therefore an integral can be zero even when there is visible space between a curve and the axis. If a question asks for area, sketch first and split at every root. Then make each region positive, either by reversing limits or by integrating the negative of a below-axis function. In the June 2025 report, AQA highlighted incorrect limits, sign errors and failure to split regions as common reasons for lost marks. 3

2. Standard antiderivatives: reverse differentiation accurately

For (n
e -1),

[\int x^n,dx=\frac{x^{n+1}}{n+1}+C.]

The power increases by one and then you divide by that new power. This is why (\int x^{-2},dx=-x^{-1}+C), not (-2x^{-3}). Constant multiples remain attached: (\int 7x^3,dx=\frac{7}{4}x^4+C). Integrate each term separately only after simplifying indices, fractions and surds.

The special logarithmic case is (\int \frac1x,dx=\ln|x|+C). Also learn the reverse derivatives (\int e^{kx},dx=\frac1k e^{kx}+C), (\int \sin(kx),dx=-\frac1k\cos(kx)+C), and (\int \cos(kx),dx=\frac1k\sin(kx)+C). A quick differentiation check is powerful: differentiate your answer mentally before moving on. For an indefinite integral, the final (+C) is a mark-protecting habit. AQA’s 2025 examiner report specifically noted candidates who integrated correctly but omitted the arbitrary constant. 3

3. Definite integrals, area under a curve and area between curves

For area between curves, find intersections first. They are the limits and they reveal which graph is upper on each interval. The working must show

[\text{Area}=\int_a^b\bigl(\text{upper function}-\text{lower function}\bigr),dx.]

A sketch earns no standalone marks in most questions, but it prevents a costly negative area. Do not assume a named function is always upper. Test one point in every interval between intersections. If the curves exchange position, split the integral. In an exam, labels such as “(y=2x) is above (y=x^2) for (0<x<2)” make your mathematical decision visible to the examiner.

4. Integration as the limit of a sum

A definite integral is the exact value approached by adding infinitely many very thin rectangles:

[\int_a^b f(x),dx=\lim_{n\to\infty}\sum_{r=1}^{n} f(x_r),\Delta x.]

For a uniform partition, (\Delta x=\frac{b-a}{n}). A finite sum is an approximation; the limit removes the approximation. In an AQA question, first identify the width, write the rectangle height in terms of (r), simplify the sigma expression, use the supplied or memorised sum formulae, and only then take the limit. Keep brackets around the whole summand. This topic is a bridge to numerical methods: a trapezium-rule estimate uses finitely many strips, whereas integration evaluates the limiting total exactly when an antiderivative is available.

5. Choosing a technique: diagnose before calculating

Most integration mistakes begin before the first line of algebra. Use the structure of the integrand as your cue.

• A straightforward sum of powers, exponentials or trig functions needs standard integration.
• A composite function with its derivative, or a constant multiple of it, suggests substitution.
• A product where one factor becomes simpler when differentiated suggests integration by parts.
• A rational expression whose denominator factorises into linear factors suggests partial fractions first.
• A differential equation with a product of an (x)-expression and a (y)-expression suggests separation of variables.

This is an exam decision, not an acronym contest. Credit is given for a valid method even if later arithmetic fails, but only when the method is set out clearly.

6. Substitution and integration by parts

For substitution, choose (u=g(x)) so that both the integrand and (dx) can be rewritten wholly in (u). Write (u=\dots), then (\frac{du}{dx}$=\dots), rearrange for the required piece and substitute every occurrence of (x). For a definite integral, change the limits into (u)-limits immediately or change back to (x) before evaluating, but never mix systems. AQA limits this to cases where one suitable substitution produces an integrable function. 1$

For parts, use the inverse product rule:

[\int u\frac{dv}{dx},dx=uv-\int v\frac{du}{dx},dx.]

Make a four-line choice box: (u), (\frac{du}{dx}), (\frac{dv}{dx}$), (v). Choose (u) to become simpler when differentiated, normally a polynomial before an exponential or trigonometric factor. The memory phrase is “UV minus VDU”. In a definite integral, retain the limits throughout and combine the completed terms into one bracket before substituting the limits. This reduces sign errors. AQA permits more than one use of parts but excludes reduction formulae. 1$

7. Partial fractions and separable differential equations

If a rational function is improper, divide first. Then factor the denominator. For distinct linear factors, write

[\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}.]

Find (A) and (B) by equating coefficients or substituting convenient values after multiplying through. Only then integrate to logarithms, using absolute values: (\int\frac{1}{x-a},dx=\ln|x-a|+C). AQA H6 specifies linear denominators, so do not spend revision time on unsupported quadratic-factor methods.

For a separable equation, move every (y)-term with (dy) and every (x)-term with (dx):

[\frac{1}{g(y)},dy=f(x),dx.]

Integrate both sides, include one constant, then use the initial condition before presenting the particular solution. Finally, interpret it. A model may be mathematically defined beyond the situation it represents: for example, time cannot be negative, a population model may become unrealistic, or a velocity expression may only apply until an object hits the ground. That contextual limitation is explicitly part of H8.

Mathematical relationships and formula status

AQA states that candidates must be able to use its Appendix B formulae and identities without them being supplied, except where they are a starting point for a proof or a result to prove. Treat every relationship below as Must memorise, not as a formula-booklet rescue. 4

Graph, data and unit skills

$Integration questions reward a purposeful sketch. Draw axes, mark every intercept or intersection used as a limit, and shade the intended region. The sketch is your sign check: a region below the axis must be treated differently when the command word is area rather than evaluate the integral. On calculator questions, use graphing or numerical integration only as a check after writing the exact method. A decimal screen value is not a replacement for a valid setup, and candidates should keep exact values such as (\frac43), (\pi) and (\ln 2) until the question asks for a decimal.$

There are no Foundation/Higher tiers in A-level Mathematics. Integration itself contains no special unit conversions, but applied questions do. In kinematics, if acceleration is in (\mathrm{m,s^{-2}}), integrating with respect to seconds gives velocity in (\mathrm{m,s^{-1}}); integrating velocity gives displacement in metres. Write a unit beside a final physical area, displacement or accumulated quantity. Units make the meaning of a constant visible and can expose an implausible answer.

Practical applications and modelling

In mechanics, integrate acceleration to obtain velocity, then integrate velocity to obtain displacement. The constant represents a physical initial velocity or position, so dropping it changes the model. In economics, a marginal cost can be integrated to recover total cost up to a fixed constant. In science, a rate function accumulates quantity. The exam skill is the same in each case: name the units, use the condition, and state a limitation rather than pretending a model works for all inputs.

Listen and recall

After listening, close this guide and answer the quick-fire questions from the podcast without writing formulas first. Retrieval, then correction, is much stronger revision than reading the worked solutions again.

References