Integration as the reverse of differentiation, indefinite and definite integrals, the area under a curve, integration of standard functions, integration by substitution and by parts, and using partial fractions to integrate rational functions.

What this dot point is asking

AQA wants you to integrate as the reverse of differentiation, evaluate indefinite and definite integrals, find areas under and between curves, integrate standard functions, and use the techniques of substitution, integration by parts and partial fractions. Definite integrals for area, and the choice of technique for a given integrand, are the recurring exam skills.

Indefinite and definite integrals

The area between a curve and the $x$-axis from $x = a$ to $x = b$ is the definite integral, taking care with regions below the axis, which give negative contributions. For total (geometric) area when the curve crosses the axis, integrate each piece separately and add the magnitudes. The area between two curves is the integral of the difference, top curve minus bottom curve.

Standard integrals

You should know $\int e^x\,dx = e^x + c$, $\int e^{kx}\,dx = \frac{1}{k}e^{kx} + c$, $\int \cos x\,dx = \sin x + c$ and $\int \sin x\,dx = -\cos x + c$. Recognising a standard form, possibly after a small adjustment for a constant inside the function, often avoids a full substitution.

Substitution

Substitution reverses the chain rule. You choose a new variable $u$ equal to an inner function, replace $dx$ using $du = \frac{du}{dx}\,dx$, and, for a definite integral, change the limits to values of $u$.

Integration by parts

For products such as $x\cos x$ or $x e^x$, parts with $u = x$ reduces the power of $x$ to a constant on the second pass. For $\int \ln x\,dx$, take $u = \ln x$ and $\frac{dv}{dx} = 1$.

Integrating with partial fractions

A rational function can be split with partial fractions first, then each term integrates to a logarithm, typically $\int \frac{1}{x - a}\,dx = \ln|x - a| + c$. This connects directly to the partial fractions work in algebra and functions, and is a common multi-step Paper 1 question.

Choosing the right technique

The hardest part of an integration question is often deciding which method applies, so it helps to recognise the signatures. If the integrand is a standard form or a small adjustment of one (a constant inside a function), integrate directly. If it contains a function and (a multiple of) its own derivative, for example $2x(x^2 + 1)^3$ or $\frac{f'(x)}{f(x)}$, use substitution; the latter integrates to $\ln|f(x)|$. If it is a product of two different kinds of function, such as a power of $x$ times an exponential or trigonometric function, use integration by parts, choosing $u$ to be the part that simplifies on differentiating. If it is a rational function (a ratio of polynomials) with a factorisable denominator, split it into partial fractions and integrate each piece to a logarithm.

A reliable definite-integral routine is: find the antiderivative first (omitting the constant), write it in square brackets with the limits, substitute the upper then the lower limit, and subtract. For an area that crosses the $x$-axis, find the roots, integrate over each interval separately, and add the magnitudes so that the regions below the axis are not subtracted away.