Knowledge organisers

Integration: A Level Maths knowledge organiser

Everything to know about integration on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Indefinite integral: The reverse of differentiation; add a constant c.
Definite integral: ∫ₐᵇ f(x) dx gives the signed area under the curve from a to b.
Integration by substitution: Change the variable to u = g(x) to make the integral simpler; change the limits too.
Integration by parts: ∫u dv/dx dx = uv − ∫v du/dx dx, for products.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Standard	\(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+c,\) \(\int\tfrac1x\,dx=\ln\|x\|+c,\) \(\int e^{kx}\,dx=\tfrac1ke^{kx}+c\)
Trig	\(\int\cos kx\,dx=\tfrac1k\sin kx+c,\) \(\int\sin kx\,dx=-\tfrac1k\cos kx+c\)
More trigIn the formula booklet: Edexcel	\(\int\sec^2kx\,dx=\tfrac1k\tan kx+c,\) \(\int\tan kx\,dx=\tfrac1k\ln\|\sec kx\|+c,\) \(\int\cot kx\,dx=\tfrac1k\ln\|\sin kx\|+c\)
Reverse chainIn the formula booklet: OCR	\(\int\frac{f'(x)}{f(x)}\,dx=\ln\|f(x)\|+c,\) \(\int f'(x)[f(x)]^n\,dx=\frac{[f(x)]^{n+1}}{n+1}+c\)
Linear inside	\(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+c\)
By partsIn the formula booklet: Edexcel, AQA, OCR	\(\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx\)
Substitution: \(u=g(x)\), replace \(dx\) by \(\frac{du}{g'(x)}\), change the limits
Area	\(\int_a^by\,dx;\) \(\text{between curves }\int_a^b(y_\text{top}-y_\text{bottom})\,dx;\) \(\text{parametric }\int y\tfrac{dx}{dt}\,dt\)
Identities to integrate	\(\sin^2x=\tfrac12(1-\cos2x),\) \(\cos^2x=\tfrac12(1+\cos2x)\)
Separable DE	\(\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)\,dx\)

Worked example

Evaluate ∫₁³ (2x + 1/x) dx, giving an exact answer.

∫(2x + 1/x) dx = x² + ln x
[x² + ln x]₁³ = (9 + ln 3) − (1 + 0)

Answer: 8 + ln 3

Common mistakes

Modelling: model not written out, limitation not in context
Integration by parts: sign errors
Constraint not used to reduce to one variable
+ c omitted

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Integrate xⁿ for n ≠ −1, including terms that must first be rewritten as powers of x, and understand integration as the reverse of differentiation.
Use definite integration to find the area between a curve and the x-axis over a given interval, sketching the region first to check it.
Integrate standard functions including xⁿ, eˣ, 1/x, sin x, cos x and sec²x, and use them to evaluate definite integrals.
Integrate by substitution, choosing a substitution such as u = 2x + 1, changing the limits for definite integrals and simplifying before integrating.
Find areas between curves, and areas under curves defined parametrically, using the integration techniques of this unit and sketches to set up the integrals.
See a definite integral as the limit of a sum of rectangle areas, and practise mixed integration questions that require choosing the right technique.

The printable sheet

Integration knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Integration knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers