A-Level Edexcel Integration Methods Reference Cheat Sheet

This cheat sheet covers the main integration methods needed for A-Level Edexcel Calculus, including substitution, integration by parts, partial fractions, and definite integrals. Students need these methods because many exam questions cannot be solved by reversing differentiation directly. A clear reference helps students choose an efficient method and avoid common algebra errors.

It is especially useful for revision, exam practice, and checking the structure of longer solutions.

The core idea is to transform a difficult integral into a simpler one using an appropriate technique. Substitution changes variables using $u = g(x)$ and $du = g'(x)\,dx$ , while integration by parts uses $\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$ . Partial fractions rewrite rational functions before integrating, often producing logarithms.

Definite integrals require limits, signed area, and sometimes a change of limits when substitution is used.

Key Facts

The basic antiderivative rule is $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ .
The logarithmic integral is $\int \frac{1}{x}\,dx = \ln{|x|} + C$ .
For substitution, if $u = g(x)$ and $du = g'(x)\,dx$ , then $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ .
Integration by parts is $\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx$ .
For definite integrals, $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F'(x) = f(x)$ .
When using substitution in a definite integral, change the limits using $u = g(a)$ and $u = g(b)$ , or convert back to $x$ before substituting limits.
A proper rational function can often be integrated after partial fractions, such as $\frac{A}{x-a} + \frac{B}{x-b}$ .
Useful trigonometric integrals include $\int \cos{x}\,dx = \sin{x} + C$ and $\int \sin{x}\,dx = -\cos{x} + C$ .

Vocabulary

Antiderivative: An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$ .
Constant of integration: The constant $C$ represents all possible vertical shifts of an indefinite integral.
Substitution: Substitution rewrites an integral using a new variable such as $u = g(x)$ to make the integrand simpler.
Integration by parts: Integration by parts is a method based on the product rule, written as $\int u\,dv = uv - \int v\,du$ .
Partial fractions: Partial fractions split a rational expression into simpler fractions that are easier to integrate.
Definite integral: A definite integral $\int_a^b f(x)\,dx$ gives the signed area between the curve and the $x$ -axis from $x=a$ to $x=b$ .

Common Mistakes to Avoid

Forgetting $+C$ in an indefinite integral is wrong because antiderivatives differ by a constant, so the full answer must include $C$ .
Using $\int x^{-1}\,dx = \frac{x^0}{0}$ is wrong because the power rule does not apply when $n = -1$ ; instead use $\int \frac{1}{x}\,dx = \ln{|x|} + C$ .
Choosing $u$ poorly in integration by parts is wrong because it can make $\int v\,du$ harder than the original integral; choose $u$ so it simplifies when differentiated.
Not changing the limits during substitution in a definite integral is wrong because the new variable has different endpoint values.
Dropping absolute value signs in logarithmic answers is wrong for expressions that can be negative, so $\int \frac{1}{x-a}\,dx = \ln{|x-a|} + C$ .

Practice Questions

1 Evaluate $\int 6x(3x^2+5)^4\,dx$ using substitution.
2 Evaluate $\int x e^{2x}\,dx$ using integration by parts.
3 Evaluate $\int_1^3 \frac{2x}{x^2+4}\,dx$ .
4 Explain how you would decide whether to use substitution, integration by parts, or partial fractions for a given integral.

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A-Level Edexcel Integration Methods Reference Cheat Sheet

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