Edexcel IAL P1 Integration: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P1 integration is the reverse of differentiation: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c$ for $n \neq - 1$ , one term at a time. Rewrite roots and fractions as powers first, and always add the constant $c$ . Given a point on the curve, substitute it to find $c$ . Definite integrals and areas come in P2.

P1 · Pure Mathematics 1
Calculator allowed
3 worked examples
32 practice questions
Tested in 4 of the last 4 papers

Practise P1 integration →

Notes
Formulas
Worked examples
Mistakes
FAQ

P1 Integration notes

Integrating powers of x

Add one to the power and divide by the new power: $\int 6 x^{2} d x = 2 x^{3} + c$ . Rewrite first: $\int (6 x^{2} + \frac{4}{x^{3}}) d x = \int (6 x^{2} + 4 x^{- 3}) d x = 2 x^{3} - 2 x^{- 2} + c$ , and $\int \sqrt{x} d x = \frac{2}{3} x^{\frac{3}{2}} + c$ . Expand brackets and split fractions before integrating; there is no rule for integrating a product term by term.

The constant of integration

Every function with the same derivative differs only by a constant, so an indefinite integral needs $+ c$ . If a curve has $f^{'} (x) = 3 x^{2} - 4$ and passes through $(2, 5)$ , then $f (x) = x^{3} - 4 x + c$ and $5 = 8 - 8 + c$ , so $c = 5$ and $f (x) = x^{3} - 4 x + 5$ .

Checking an answer

Differentiate your answer: you should get back exactly the function you integrated. This catches a forgotten division or a sign slip with negative powers.

Key formulas

Differentiate / integrate $x^{n}$ Learn it: $\frac{d}{d x} x^{n} = n x^{n - 1}, \int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c (n \neq - 1)$

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 P1 formulas on one page (free PDF) →

P1 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

Given that $f^{″} (x) = 12 x - 4$ , $f^{'} (1) = 2$ and $f (- 1) = 5$ , find $f (x)$ .

Mark scheme

M1 for integrating $f^{″} (x)$ to find $f^{'} (x) = 6 x^{2} - 4 x + c_{1}$ .
M1 for substituting $f^{'} (1) = 2$ to find $c_{1}$ : $2 = 6 (1)^{2} - 4 (1) + c_{1} ⟹ c_{1} = 0$ .
M1 for substituting $c_{1} = 0$ and integrating again to find $f (x) = 2 x^{3} - 2 x^{2} + c_{2}$ .
M1 for substituting $f (- 1) = 5$ : $5 = 2 (- 1)^{3} - 2 (- 1)^{2} + c_{2}$ .
M1 for solving for $c_{2}$ : $5 = - 2 - 2 + c_{2} ⟹ c_{2} = 9$ .
A1 for the final correct function $f (x) = 2 x^{3} - 2 x^{2} + 9$ .

6 steps · 1 to fill in

Worked example 2 · 5 marks · hard

The gradient of a curve is given by $\frac{d y}{d x} = 4 \sqrt{x} - 1$ . The curve passes through the point $(9, 20)$ . Find the equation of the curve.

Mark scheme

M1 for rewriting as $4 x^{\frac{1}{2}} - 1$ and integrating: $y = \frac{4 x^{\frac{3}{2}}}{\frac{3}{2}} - x + c$ .
A1 for simplifying the integral to $y = \frac{8}{3} x^{\frac{3}{2}} - x + c$ .
M1 for substituting the point $(9, 20)$ : $20 = \frac{8}{3} (9)^{\frac{3}{2}} - 9 + c$ .
M1 for evaluating $9^{\frac{3}{2}} = 27$ to solve for $c$ : $20 = 72 - 9 + c ⟹ c = - 43$ .
A1 for the final equation $y = \frac{8}{3} x^{\frac{3}{2}} - x - 43$ .

5 steps · 1 to fill in

Worked example 3 · 5 marks · hard

Given that $f^{'} (x) = \frac{6 x^{3} + 1}{x^{2}}$ and $f (1) = 4$ , find an expression for $f (x)$ .

Mark scheme

M1 for splitting the fraction: $f^{'} (x) = 6 x + x^{- 2}$ .
M1 for integrating to find $f (x)$ : $3 x^{2} + \frac{x^{- 1}}{- 1} + c$ .
A1 for simplifying to $f (x) = 3 x^{2} - x^{- 1} + c$ .
M1 for substituting $(1, 4)$ to find $c$ : $4 = 3 (1)^{2} - (1)^{- 1} + c ⟹ 4 = 2 + c ⟹ c = 2$ .
A1 for the final function $f (x) = 3 x^{2} - \frac{1}{x} + 2$ .

5 steps · 1 to fill in

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Common P1 integration mistakes

No + c. An indefinite integral without + c loses a mark, and a curve question cannot be finished without it.
Integrating a product term by term. $\int x (x + 2) d x$ is not $\frac{x^{2}}{2} (\frac{x^{2}}{2} + 2 x)$ : expand to $x^{2} + 2 x$ first.
Negative powers. $\int x^{- 3} d x = \frac{x^{- 2}}{- 2} + c$ : divide by the new power, $- 2$ , not by $- 3$ .
Dividing by the old power. $\int x^{4} d x = \frac{x^{5}}{5} + c$ , not $\frac{x^{5}}{4}$ .

Every P1 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.

P1 Integration FAQ

Why do I add + c when I integrate?

Differentiating a constant gives zero, so many functions share the same derivative. The + c stands for all of them; a point on the curve tells you which one you want.

Is definite integration in IAL P1?

No. P1 covers indefinite integration of powers of x and finding the constant from a point. Definite integrals, areas and the trapezium rule are in P2.

Is a calculator allowed in IAL P1?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P1 included. Method marks are for the working you write down, so show it.

How long is the IAL P1 exam?

Each IAL maths unit, P1 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does integration come up in P1?

In our mapping of the 4 Pearson P1 papers from June 2024 to June 2025, integration was tested in 4 of them (7 questions in all). Any topic can be examined in any series.

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More P1: Algebra · Coordinate Geometry · Trigonometry · Differentiation · Graphs and Transformations

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Other units: IAL P2 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P1 Integration: exam questions and answers, notes and worked examples

Try it first

P1 Integration notes

Integrating powers of x

The constant of integration

Checking an answer

Key formulas

P1 Integration worked examples

Worked example 1 · 6 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 5 marks · hard

Mark scheme

Common P1 integration mistakes

P1 Integration FAQ

Why do I add + c when I integrate?

Is definite integration in IAL P1?

Is a calculator allowed in IAL P1?

How long is the IAL P1 exam?

How often does integration come up in P1?

Next steps

Your P1 revision kit