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A-Level Edexcel Integration Methods Reference cheat sheet - grade 11-12

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Calculus Grade 11-12

A-Level Edexcel Integration Methods Reference Cheat Sheet

A printable reference covering substitution, integration by parts, partial fractions, trigonometric integrals, and definite integration for grades 11-12.

TopicsIntegralsTrigonometryAlgebraFunctionsCalculusStudy Guide
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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)u = g(x) and du=g′(x) dxdu = g'(x)\,dx, while integration by parts uses ∫u dvdx dx=uv−∫v dudx dx\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 ∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C for n≠−1n \neq -1.
  • The logarithmic integral is ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx = \ln{|x|} + C.
  • For substitution, if u=g(x)u = g(x) and du=g′(x) dxdu = g'(x)\,dx, then ∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x)\,dx = \int f(u)\,du.
  • Integration by parts is ∫u dvdx dx=uv−∫v dudx dx\int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx.
  • For definite integrals, ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a) where F′(x)=f(x)F'(x) = f(x).
  • When using substitution in a definite integral, change the limits using u=g(a)u = g(a) and u=g(b)u = g(b), or convert back to xx before substituting limits.
  • A proper rational function can often be integrated after partial fractions, such as Ax−a+Bx−b\frac{A}{x-a} + \frac{B}{x-b}.
  • Useful trigonometric integrals include ∫cos⁡x dx=sin⁡x+C\int \cos{x}\,dx = \sin{x} + C and ∫sin⁡x dx=−cos⁡x+C\int \sin{x}\,dx = -\cos{x} + C.

Vocabulary

Antiderivative
An antiderivative of f(x)f(x) is a function F(x)F(x) such that F′(x)=f(x)F'(x) = f(x).
Constant of integration
The constant CC represents all possible vertical shifts of an indefinite integral.
Substitution
Substitution rewrites an integral using a new variable such as u=g(x)u = g(x) to make the integrand simpler.
Integration by parts
Integration by parts is a method based on the product rule, written as ∫u dv=uv−∫v du\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 ∫abf(x) dx\int_a^b f(x)\,dx gives the signed area between the curve and the xx-axis from x=ax=a to x=bx=b.

Common Mistakes to Avoid

  • Forgetting +C+C in an indefinite integral is wrong because antiderivatives differ by a constant, so the full answer must include CC.
  • Using ∫x−1 dx=x00\int x^{-1}\,dx = \frac{x^0}{0} is wrong because the power rule does not apply when n=−1n = -1; instead use ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx = \ln{|x|} + C.
  • Choosing uu poorly in integration by parts is wrong because it can make ∫v du\int v\,du harder than the original integral; choose uu 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 ∫1x−a dx=ln⁡∣x−a∣+C\int \frac{1}{x-a}\,dx = \ln{|x-a|} + C.

Practice Questions

  1. 1 Evaluate ∫6x(3x2+5)4 dx\int 6x(3x^2+5)^4\,dx using substitution.
  2. 2 Evaluate ∫xe2x dx\int x e^{2x}\,dx using integration by parts.
  3. 3 Evaluate ∫132xx2+4 dx\int_1^3 \frac{2x}{x^2+4}\,dx.
  4. 4 Explain how you would decide whether to use substitution, integration by parts, or partial fractions for a given integral.

Understanding A-Level Edexcel Integration Methods Reference

Choosing a method is usually the hardest part. Look first at the structure, not at every term separately. A function inside another function often points to substitution.

The derivative of the inner function should be present, perhaps multiplied by a constant. For example, an expression involving x times cosine of x squared can be changed by using x squared as the new variable. The remaining x factor supplies part of the required derivative.

Constants can be moved outside before starting. This habit prevents a common mistake where students choose a substitution but leave some x terms mixed with the new variable.

Integration by parts is most useful for a product where one factor becomes simpler when differentiated. Polynomials usually become shorter each time they are differentiated. Logarithms and inverse trigonometric functions are often good choices for the part called u because their derivatives are manageable.

Exponentials and ordinary trigonometric functions are often suitable for the part that is integrated first. Check the result by differentiating the two terms produced at the end. The minus sign matters.

In repeated integration by parts, a sign error near the beginning can change every later term. Some products return to the original integral after two applications. In that case, collect the original integral on one side before solving for it.

Partial fractions are an algebra task before they become an integration task. First make sure the fraction is proper. If the degree of the numerator is at least the degree of the denominator, use division first.

Then factor the denominator as far as possible. A repeated linear factor needs a separate fraction for every power up to that repeated power. A quadratic factor that cannot be factorised needs a linear numerator above it.

Students often lose marks by writing an incomplete set of fractions, even when their later integration is accurate. Trigonometric integrals need similar preparation.

Identities can turn squared sine or squared cosine terms into simpler forms. For powers of sine and cosine, an odd power can often be saved to form a derivative, while even powers often need a half angle identity.

Definite integration measures net accumulation over an interval. Areas below the horizontal axis count negatively, so a definite integral is not always the same as total area. For total area, find where the graph crosses the axis and split the interval there.

This idea appears in displacement from velocity, work done by a changing force, and accumulated probability from a density curve. Keep exact values for as long as possible, especially when logarithms or trigonometric values occur. When substitution is used with limits, use one variable system throughout.

Changing limits to the new variable is usually cleaner. Final checks are powerful. Differentiate an indefinite answer, inspect whether logarithm inputs have valid domains, and decide whether the sign and size of a definite answer fit the graph.