Knowledge organisers

Pure Mathematics 4 (P4): Edexcel IAL Maths knowledge organiser

Everything to know about pure mathematics 4 (p4) 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

Partial fractions: Splitting a fraction such as 1/((x − 1)(x + 2)) into A/(x − 1) + B/(x + 2).
Implicit differentiation: Differentiating each term with respect to x; a term in y picks up a factor dy/dx.
Parametric equations: x and y written in terms of t; dy/dx = (dy/dt)/(dx/dt).
Vector equation of a line: r = a + λb through the point with position vector a, in direction b.

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.

Binomial, \(\|x\|<1\), \(n\in\mathbb Q\)In the formula booklet	\((1+x)^n=1+nx+\frac{n(n-1)}{1\times2}x^2+\cdots\)
Partial fractions	\(\frac{\dots}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}=\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}\)
Parametric; implicit	\(\frac{dy}{dx}=\frac{dy/dt}{dx/dt};\) \(\tfrac{d}{dx}y^2=2y\tfrac{dy}{dx}\)
IntegralsIn the formula booklet	\(\int\sec^2kx\,dx=\tfrac1k\tan kx,\) \(\int\tan x\,dx=\ln\|\sec x\|,\) \(\int\cot x\,dx=\ln\|\sin x\|\)
More integralsIn the formula booklet	\(\int\cosec x\,dx=-\ln\|\cosec x+\cot x\|,\) \(\int\sec x\,dx=\ln\|\sec x+\tan x\|\)
By partsIn the formula booklet	\(\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx\)
Reverse chain	\(\int\frac{f'(x)}{f(x)}\,dx=\ln\|f(x)\|+c\)
Volume of revolution	\(V=\pi\int_a^by^2\,dx\)
Separable DE	\(\int\frac{dy}{g(y)}=\int f(x)\,dx\)
Vectors	\(\mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3=\|\mathbf a\|\|\mathbf b\|\cos\theta\)
Line; perpendicular	\(\mathbf r=\mathbf a+\lambda\mathbf b;\) \(\mathbf a\cdot\mathbf b=0\)
Magnitude	\(\|\mathbf a\|=\sqrt{a_1^2+a_2^2+a_3^2}\)

Worked example

Find the exact value of ∫₀¹ x e^(2x) dx.

By parts: u = x, dv/dx = e^(2x), so v = ½e^(2x)
∫ x e^(2x) dx = ½x e^(2x) − ¼e^(2x)
[½x e^(2x) − ¼e^(2x)]₀¹ = (½e² − ¼e²) − (−¼)

Answer: (e² + 1)/4

Common mistakes

Differential equations: the constant added too late
Binomial (any index): factor not raised to the power
Assuming the statement to be proved rather than its negation
A/(x+1)² given without the A/(x+1) term

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

You should be able to…

Write proofs by contradiction, stating the opposite assumption clearly and showing how it leads to an impossible result, such as there being no largest even number.
Model paths such as a ball's flight or a moving point with parametric equations, and interpret the parameter and the answers in context.
Use the chain rule to connect rates of change, such as finding how fast the volume of a balloon changes as its radius grows.
Choose an efficient integration method, from recognition, substitution, parts, partial fractions and trigonometric identities, for a mixed set of integrals.
Represent vectors in three dimensions with i, j and k, find their magnitudes and unit vectors, and find angles between a vector and the axes.
Solve multi-step vector geometry problems in 3D, such as finding the area of a triangle or a reflection of a point in a line.

The printable sheet

Pure Mathematics 4 (P4) knowledge organiser for Edexcel IAL Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Pure Mathematics 4 (P4) knowledge organiser (Edexcel IAL Maths), A4. Download the PDF.

Revise it next

Other Edexcel IAL Maths topics: Pure Mathematics 1 (P1) · Pure Mathematics 2 (P2) · Pure Mathematics 3 (P3) · Statistics 1 (S1) · Mechanics 1 (M1) · All Edexcel IAL Maths organisers