Find the exact value of $\int_{0}^{\pi} x^2 \sin x \, dx$.

M1 for setting up the first integration by parts with $u = x^2$ and $dv = \sin x \, dx$. M1 for obtaining $-x^2 \cos x + \int 2x \cos x \, dx$. M1 for applying integration by parts a second time on $\int 2x \cos x \, dx$ with $u=2x$ and $dv=\cos x \, dx$. A1 for the fully integrated expression $-x^2 \cos x + 2x \sin x + 2 \cos x$. M1 for substituting the limits $0$ and $\pi$: $(-\pi^2(-1) + 0 + 2(-1)) - (0 + 0 + 2)$. A1 for calculating the exact final answer $\pi^2 - 4$.

A Level Math Revision

Edexcel IAL P4 Integration: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P4 integration adds the techniques: recognising $\int \frac{f^{'} (x)}{f (x)} d x = \ln | f (x) | + c$ , integration by substitution, integration by parts, partial fractions, trig identities, and volumes of revolution about the $x$ -axis, $V = π \int y^{2} d x$ . The skill is choosing the method: look at the integrand's structure before you start.

P4 · Pure Mathematics 4
Calculator allowed
3 worked examples
30 practice questions
Tested in 4 of the last 4 papers

Practise P4 integration →

Notes
Formulas
Worked examples
Mistakes
FAQ

P4 Integration notes

Standard results and recognition

If the numerator is (a multiple of) the derivative of the denominator, the answer is a log: $\int \frac{2 x}{x^{2} + 1} d x = \ln (x^{2} + 1) + c$ . Likewise $\int \sec^{2} x d x = \tan x + c$ .

Integration by substitution

Choose $u$ , find $\frac{d u}{d x}$ , and replace every $x$ and the $d x$ . With $u = x^{2} + 3$ : $\int x (x^{2} + 3)^{4} d x = \frac{(x^{2} + 3)^{5}}{10} + c$ . For a definite integral, change the limits to $u$ -values too.

Integration by parts

$\int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x$ . Let $u$ be the factor that gets simpler when differentiated (a power of $x$ ), unless there is a $\ln x$ , which should be $u$ . So $\int x \cos x d x = x \sin x + \cos x + c$ .

Using partial fractions

Split the fraction first, then integrate each part as a log: $\int \frac{3}{(x - 1) (x + 2)} d x = \ln | x - 1 | - \ln | x + 2 | + c$ .

Using trig identities

$\sin^{2} x$ and $\cos^{2} x$ cannot be integrated directly: use $\cos 2 x = 1 - 2 \sin^{2} x$ to get $\int \sin^{2} x d x = \frac{x}{2} - \frac{\sin 2 x}{4} + c$ .

Volumes of revolution

Rotating $y = f (x)$ through $2 π$ about the $x$ -axis between $x = a$ and $x = b$ gives $V = π \int_{a}^{b} y^{2} d x$ . For $y = \sqrt{x}$ from 0 to 4, $V = π \int_{0}^{4} x d x = 8 π$ .

Key formulas

By parts In the formula booklet: $\int u \frac{d v}{d x} d x = u v - \int v \frac{d u}{d x} d x$
Integrals In the formula booklet: $\int \sec^{2} k x d x = \frac{1}{k} \tan k x, \int \tan x d x = \ln | \sec x |, \int \cot x d x = \ln | \sin x |$
Reverse chain Learn it: $\int \frac{f^{'} (x)}{f (x)} d x = \ln | f (x) | + c$
Partial fractions Learn it: $\frac{\dots}{(x - a) (x - b)} = \frac{A}{x - a} + \frac{B}{x - b}, \frac{\dots}{(x - a)^{2} (x - b)} = \frac{A}{x - a} + \frac{B}{(x - a)^{2}} + \frac{C}{x - b}$
Volume of revolution Learn it: $V = π \int_{a}^{b} y^{2} d x$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P4 formulas on one page (free PDF) →

P4 Integration worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 6 marks · hard

Using the substitution $u^{2} = 2 x - 1$ , find $\int x \sqrt{2 x - 1} d x$ .

Mark scheme

M1 for implicit differentiation $2 u d u = 2 d x ⟹ d x = u d u$ .
M1 for rearranging the substitution to find $x = \frac{u^{2} + 1}{2}$ .
M1 for substituting everything into the integral: $\int (\frac{u^{2} + 1}{2}) (u) (u) d u$ .
M1 for expanding to $\frac{1}{2} \int (u^{4} + u^{2}) d u$ .
M1 for integrating to $\frac{1}{10} u^{5} + \frac{1}{6} u^{3}$ .
A1 for replacing $u$ with $(2 x - 1)^{\frac{1}{2}}$ to get $\frac{1}{10} (2 x - 1)^{\frac{5}{2}} + \frac{1}{6} (2 x - 1)^{\frac{3}{2}} + c$ .

6 steps

Worked example 2 · 5 marks · hard

Find $\int \frac{x^{2} + 1}{x (x - 1)} d x$ .

Mark scheme

M1 for recognising the improper fraction and performing division to get $1 + \frac{x + 1}{x (x - 1)}$ .
M1 for setting up partial fractions for the remainder: $\frac{x + 1}{x (x - 1)} = \frac{A}{x} + \frac{B}{x - 1}$ .
A1 for determining the constants $A = - 1$ and $B = 2$ .
M1 for formulating the complete integral: $\int (1 - \frac{1}{x} + \frac{2}{x - 1}) d x$ .
A1 for integrating the terms correctly to $x - \ln | x | + 2 \ln | x - 1 | + c$ .

5 steps · 1 to fill in

Worked example 3 · 6 marks · hard

Find the exact value of $\int_{0}^{π} x^{2} \sin x d x$ .

Mark scheme

M1 for setting up the first integration by parts with $u = x^{2}$ and $d v = \sin x d x$ .
M1 for obtaining $- x^{2} \cos x + \int 2 x \cos x d x$ .
M1 for applying integration by parts a second time on $\int 2 x \cos x d x$ with $u = 2 x$ and $d v = \cos x d x$ .
A1 for the fully integrated expression $- x^{2} \cos x + 2 x \sin x + 2 \cos x$ .
M1 for substituting the limits $0$ and $π$ : $(- π^{2} (- 1) + 0 + 2 (- 1)) - (0 + 0 + 2)$ .
A1 for calculating the exact final answer $π^{2} - 4$ .

6 steps

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Common P4 integration mistakes

Half a substitution. Every $x$ , including the $d x$ , must go. A mix of $x$ and $u$ under one integral sign earns little.
Old limits after substituting. In a definite integral either change the limits to u-values or substitute back before using the x-limits, never a mix.
Choosing u badly in parts. With $\int x e^{x} d x$ , $u = x$ works; $u = e^{x}$ makes the new integral harder.
Volume without the square or the π. $V = π \int y^{2} d x$ : square $y$ first and keep the $π$ .

Every P4 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find a numerical derivative and definite integral. A calculator is allowed in every IAL unit.

P4 Integration FAQ

Is integration by parts in the IAL formula booklet?

Yes: on our IAL formula sheet the integration by parts formula is marked In the formula booklet. The volume of revolution formula and ∫f′(x)/f(x) dx are marked Learn it.

How do I choose between substitution and parts?

Substitution suits a function and (a multiple of) its inner derivative, such as x(x² + 3)⁴. Parts suits a product of two unrelated types, such as x cos x or x eˣ, or a lone ln x.

Is a calculator allowed in IAL P4?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P4 included. Method marks are for the working you write down, so show it.

How long is the IAL P4 exam?

Each IAL maths unit, P4 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does integration come up in P4?

In our mapping of the 4 Pearson P4 papers from June 2024 to June 2025, integration was tested in 4 of them (11 questions in all). Any topic can be examined in any series.

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More P4: Vectors · Partial Fractions · Parametric Equations · Differential Equations · Proof · The Binomial Expansion · Differentiation

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Other units: IAL P1 · IAL P2 · IAL P3 · IAL S1 · IAL M1

Edexcel IAL P4 Integration: exam questions and answers, notes and worked examples

Try it first

P4 Integration notes

Standard results and recognition

Integration by substitution

Integration by parts

Using partial fractions

Using trig identities

Volumes of revolution

Key formulas

P4 Integration worked examples

Worked example 1 · 6 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 6 marks · hard

Mark scheme

Common P4 integration mistakes

P4 Integration FAQ

Is integration by parts in the IAL formula booklet?

How do I choose between substitution and parts?

Is a calculator allowed in IAL P4?

How long is the IAL P4 exam?

How often does integration come up in P4?

Next steps

Your P4 revision kit