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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WMA11 · Pure Mathematics 1IntegrationIntegrating powers of x

Integration · Subtopic 2 of 3

Integrating Powers of x

Integration undoes differentiation: raise the power by one and divide by the new power, then add $+c$ because any constant would have vanished when differentiating. As with differentiation, rewrite roots, fractions and brackets as separate powers of $x$ first.

Edexcel YMA01 WMA11≈ 30 minRevision 5 · updated 2026-10-27

Practise this topicTimed test Ask Salam Tutor

Section 1 of 14

Learning objectives

Understand integration as the reverse of differentiation.
Integrate $x^n$ for $n\ne-1$ , term by term.
Include the constant of integration.
Rewrite roots, fractions and products before integrating.

Pause and think

$x^2$ , $x^2+5$ and $x^2-17$ all differentiate to $2x$ . So what is "the" function whose derivative is $2x$ ?

Section 2 of 14

Core idea

Start hereUndoing differentiation

If a car's speed is known at every moment, can you recover where it is? Going backwards from a gradient to the original function is integration, and there is a twist: many functions share the same gradient.

The big idea

$\int x^n\,dx=\frac{x^{n+1}}{n+1}+c$ ( $n\neq-1$ ). Add $c$ : functions differing by a constant have the same derivative.

1. Reversing the power rule

Every curve y = x² + c has gradient 2x

c = -3.0

Sliding the curve up or down changes nothing about its steepness. That is the family

\int2x\,dx=x^2+c

Why the rule works

$\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right)=\frac{(n+1)x^n}{n+1}$
Power rule.
$=x^n$
Cancel.
$\frac{d}{dx}(c)=0$
So any constant can be added.

n=-1

fails (division by zero):

\int x^{-1}dx

needs logarithms, beyond P1.

Worked example $\int\left(4x^3-6x+\frac2{x^2}\right)dx$

Integrate.

Goal: Antiderivative.
Strategy: Powers; raise by 1, divide by new power.

$4x^3-6x+2x^{-2}$
Rewrite.
$x^4-3x^2+\frac{2x^{-1}}{-1}+c$
Integrate each.
$=x^4-3x^2-\frac2x+c$
Tidy.

Answer

x^4-3x^2-\frac2x+c

✓ CheckDifferentiate:

4x^3-6x+2x^{-2}

✓.

Worked example $\int\frac{x^2+3}{\sqrt x}dx$

Integrate.

Goal: Antiderivative.
Strategy: Split the fraction.

$x^{3/2}+3x^{-1/2}$
$\frac{x^2}{x^{1/2}}$ , $\frac3{x^{1/2}}$ .
$\tfrac25x^{5/2}+6x^{1/2}+c$
Divide by $\frac52$ and $\frac12$ .

Answer

\frac25x^{5/2}+6\sqrt x+c

✓ CheckDerivative

x^{3/2}+3x^{-1/2}

✓.

Common misconceptionIntegrating a fraction piece by piece

\int\frac{x^2+3}{\sqrt x}\neq\frac{\int(x^2+3)}{\int\sqrt x}

. Simplify first.

Pause and think

$\int6\,dx$ ?

Summary

Raise power, divide.
Always $+c$ .

Integral: Antiderivative.
Constant of integration: $c$ .

HistoryThe ∫ sign. Leibniz introduced the symbol

\int

in 1675. It is a long letter S, for the Latin summa, because he thought of integration as adding up infinitely many thin strips.

Fun factThe one power that breaks the rule. The rule fails for

x^{-1}

because it would mean dividing by

0

. The integral of

\dfrac1x

turned out to need a whole new function, the natural logarithm, which you meet later in the course.

Section 3 of 14

Key formulae

Power rule

\int x^n\,dx=\frac{x^{n+1}}{n+1}+c,\quad n\ne-1

Constants

\int k\,dx=kx+c

TrickCheck by differentiating. Differentiate your answer: you should get back exactly the expression you integrated. It takes ten seconds and catches nearly every sign and fraction slip.

Section 4 of 14

Worked examples

Example 1 · Negative power0 / 2 steps

Find

\displaystyle\int\left(3\sqrt x-\frac4{x^3}\right)dx

Example 2 · Divide first0 / 2 steps

Find

\displaystyle\int\frac{2x^3-5}{x^2}\,dx

Section 5 of 14

Explore the graph

A family of antiderivatives

Loading the interactive graph…

Try this

The parabola is $y=x^2$ . The cubic is $y=\tfrac13x^3+c$ , an antiderivative of $x^2$ .
Change $c$ . The cubic slides up and down, but its gradient at every $x$ stays the same.
Check: at $x=2$ , every cubic has gradient $4$ , the height of the parabola there.

Every curve $y=\tfrac13x^3+c$ has derivative $x^2$ .

In the real worldFrom speed to distance. If you know a car's speed at every moment, integrating gives the distance travelled. Sat-navs and fitness trackers do a version of this to estimate how far you have gone between GPS readings.

Section 6 of 14

Common mistakes

IncorrectMultiplying by the power instead of dividing

CorrectDivide by the new power

That is the reverse of differentiating.

IncorrectForgetting

+c

CorrectAlways add

+c

Indefinite integrals have a family of answers.

Incorrect

\int-4x^{-3}\,dx=-2x^{-2}

Correct

+2x^{-2}

\tfrac{-4}{-2}=+2

IncorrectIntegrating a product factor by factor

CorrectExpand first

\int(x+3)(x-1)\,dx\ne\int(x+3)\,dx\times\int(x-1)\,dx

$\int x^3\,dx$

Raise the power by one, divide by the new power, and never forget $+c$

Section 7 of 14

Exam tip

Exam tipThe +c mark. Indefinite integration questions usually reserve a final mark for "all correct, including

+c

". Write it on every indefinite integral, every time.

Section 8 of 14

Guided practice

SA-IAL-P1-IN-G012 marksL1 FoundationCalculatorPower rule

Find

\displaystyle\int(6x^2+4x)\,dx

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-P1-IN1-0011 markL1 FoundationNon-calculatorPower rule

Find

\displaystyle\int8x^3\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN1-0022 marksL1 FoundationNon-calculatorPower rule

Find

\displaystyle\int x^{-\frac12}\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN1-0032 marksL1 FoundationNon-calculatorPower rule

Find

\displaystyle\int(4-3x^2)\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN-0082 marksL2 StandardCalculatorPower rule

Find

\displaystyle\int5x^{\frac23}\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN-0023 marksL2 StandardCalculatorRewrite first

Find

\displaystyle\int(x+3)(x-1)\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN1-0042 marksL2 StandardNon-calculatorRewrite first

Find

\displaystyle\int\frac6{x^4}\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN-0013 marksL2 StandardCalculatorRewrite first

Find

\displaystyle\int\left(3\sqrt x-\frac4{x^3}\right)dx

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-P1-IN1-0053 marksL2 StandardNon-calculatorRewrite first

Find

\displaystyle\int(2x+1)^2\,dx

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-IAL-P1-IN-0034 marksL3 ExamCalculatorRewrite first

Find

\displaystyle\int\frac{2x^3-5}{x^2}\,dx

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+c$ , $n\ne-1$ .
Integrate term by term; one $+c$ .
Rewrite roots, fractions and brackets first.
Check by differentiating.

Section 12 of 14

Quick quiz

Q1 · SA-IAL-P1-IN-Q011 markL1 FoundationCalculatorPower rule

Why do we add

+\,c

to an indefinite integral?

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-IAL-P1-IN-Q021 markL1 FoundationCalculatorPower rule

\displaystyle\int x^{-2}\,dx=

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WMA11 (Pure 1)
Where: P1 · 8.1 Indefinite integration
Expected: Integrate xⁿ for rational n ≠ −1, and sums and differences of such terms; understand integration as the reverse of differentiation and include the constant of integration.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: "Find ∫(…) dx, giving each term in its simplest form." Expect roots, negative powers and fractions to be rewritten first.
Open in: YMA01 · WMA11

Section 14 of 14

Past paper questions

2 questions (7 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WMA11/01 · January 2025Question 14 marks
Integrate an expanded expression in powers of x and simplify.
Question paper, page 2 Mark scheme
WMA11/01 · June 2024Question 13 marks
Integrate powers of x.
Question paper, page 2 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).