Indefinite and definite integrals, areas under curves, integrals of standard functions, integration by substitution and by parts, integration using partial fractions, and differential equations.

What this dot point is asking

Edexcel wants you to integrate standard functions, find indefinite and definite integrals, calculate areas under and between curves, integrate by substitution and by parts, use partial fractions to integrate rational functions, and solve first-order differential equations by separating variables.

The answer

Indefinite and definite integrals

Standard integrals include $\int e^x\,dx = e^x + c$, $\int \tfrac{1}{x}\,dx = \ln|x| + c$, $\int \cos x\,dx = \sin x + c$ and $\int \sin x\,dx = -\cos x + c$.

Substitution and by parts

Substitution reverses the chain rule: replace part of the integrand with $u$ and convert $dx$ using $\dfrac{du}{dx}$. The aim is to choose $u$ so that the integral becomes a standard one in $u$. For a definite integral you also change the limits into $u$-values, which saves converting back to $x$ at the end.

Integration using partial fractions

A rational function whose denominator factorises can be split into simpler fractions, each of which integrates to a logarithm. This is the standard route for integrands such as $\dfrac{1}{(x - a)(x - b)}$.

Areas

The area between a curve and the $x$-axis is $\int_a^b y\,dx$. Where the curve dips below the axis the integral is negative, so split the region at the roots and add the magnitudes. The area between two curves $y = f(x)$ and $y = g(x)$ with $f \ge g$ is $\int_a^b (f(x) - g(x))\,dx$.

Differential equations

A separable equation $\dfrac{dy}{dx} = f(x)g(y)$ is solved by writing $\int \dfrac{1}{g(y)}\,dy = \int f(x)\,dx$ and integrating each side, then using a boundary condition to find the constant. Such equations model situations where the rate of change depends on the current value, such as cooling, population growth and the discharge of a capacitor.

Examples in context

Try this

Q1. Find $\int (6x^2 - 4x + 1)\,dx$. [2 marks]

• Cue. $2x^3 - 2x^2 + x + c$.

Q2. Evaluate $\int_1^2 \dfrac{1}{x}\,dx$. [3 marks]

• Cue. $[\ln x]_1^2 = \ln 2 - \ln 1 = \ln 2$.