Where can I learn Integration for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 3.5 (Integration) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/3-5-integration.

Is Integration (3.5) on 9709 free to study?

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3.5Mathematics22 min readPREMIUM

Integration

Welcome to integration in Pure Mathematics 3. While you're familiar with basic integration from P1, this topic introduces powerful new techniques essential for solving a much wider range of problems. We will master integration by substitution and by parts, and apply these methods to find areas, and a new concept: volumes of solids formed by revolution. These skills are fundamental not just for your exams, but for any further study in mathematics, physics, or engineering.

BY THE END, YOU CAN…

exam checklist — tick these off

1Integrate standard functions including exponential, logarithmic, and trigonometric forms.
2Apply integration by substitution and integration by parts to solve more complex integrals.
3Use definite integration to calculate the exact area under a curve or between two curves.
4Calculate the volume of a solid of revolution by rotating a region about the x-axis or y-axis.

STUDY MODE

READING

TYPEFACE

Literata — a reading serif made for e-books; set like a textbook, sits well beside maths.

SIZE

AIRY SPACING

Wider letter and line spacing. The one change with solid evidence for readers who lose their place — worth a try whoever you are.

Pick whatever reads fastest for you — it differs from person to person, by up to a third.

EXPLAIN SIMPLER

The same notes in plain English — no jargon. Key points and exam tips stay.

DIAGRAM NOTES

A small sketch beside each block of notes.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Un-doing Derivatives to Find Area

Integration is the reverse process of differentiation, allowing us to find the original function from its rate of change. We use this to calculate the exact area under curves, a fundamental concept in calculus.

THINK OF IT LIKE…

Imagine you have a speedometer reading (your speed, the derivative) for your entire car journey. Integration is like using that data to calculate the total distance you've travelled (the original function). A definite integral from 1 pm to 2 pm would tell you exactly how far you travelled in that hour.

1
Integration reverses differentiation — find the antiderivative F(x) with F′(x) = f(x).
2
A definite integral ∫ₐᵇ f(x) dx equals the signed area under the curve.
3
Evaluate F(b) − F(a) after finding F(x); always substitute limits explicitly.
4
Integration by parts: ∫ u dv = uv − ∫ v du — choose u using LIATE.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

1 more simulation for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

1 simulation

GeoGebraCore9709 3.5
Area between two curves
Drag the red points on the x-axis to set the limits, or type in your own pair of functions.
Try this
1. Drag the two red points to the x-values where the curves cross.
2. Read the shaded area, then integrate (upper − lower) by hand between the same limits.
3. Push one limit past a crossing point and watch the running total start to fall.
Look for Area = ∫(upper − lower) dx between the intersection points; where the curves swap over, the integral has to be split.
Open on GeoGebra
Ben Roth · GeoGebra · GeoGebra Terms of Service

Key formulas

Tap any symbol to reveal exactly what it means and its units.

KEY RELATION

$\int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + C \\ \int \frac{1}{ax+b} dx = \frac{1}{a}\ln|ax+b| + C \\ \int \sin(ax+b) dx = -\frac{1}{a}\cos(ax+b) + C \\ \int \sec^2(ax+b) dx = \frac{1}{a}\tan(ax+b) + C$

Tap a symbol — great for exam definitions

KEY RELATION

$\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Standard Integrals and Key Forms

In P3, we expand our library of standard integrals. The key is to recognise the function's form and apply the correct rule. Pay close attention to the effect of linear transformations of the form $(ax+b)$ inside the function.

The factor of $\frac{1}{a}$ appears when integrating a function of a linear argument $(ax+b)$ . This comes from reversing the chain rule.
Always include the constant of integration, $+C$ , for indefinite integrals.
The modulus sign in $\ln|ax+b|$ is crucial as the logarithm is only defined for positive inputs.

Not clear?

Integration by Substitution

Integration by substitution is a technique for simplifying integrals that are not in a standard form. It is essentially the reverse of the chain rule for differentiation. The goal is to choose a substitution, typically letting 'u' equal a part of the integrand, which transforms the integral into a simpler one that we know how to solve. When dealing with definite integrals, remember that the limits of integration must also be converted to the new variable.

Not clear?

Integration by Parts

Integration by parts is our method for integrating a product of two functions, derived from the product rule for differentiation. The key to success is choosing which function to label as 'u' (the one to be differentiated) and which to label as $\frac{dv}{dx}$ (the one to be integrated). A useful mnemonic is LIATE (Logarithmic, Inverse Trig, Algebraic, Trig, Exponential), which suggests the order of preference for choosing 'u'.

The Integration by Parts Formula: \ $\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$

Not clear?

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

reveal slowly — mark each step

EXAMPLE 1

Find the exact area of the region enclosed by the curve $y = x\sqrt{x^2+5}$ , the x-axis, and the lines $x=0$ and $x=2$ .

1
The area is given by the definite integral $A = \int_0^2 x\sqrt{x^2+5} \, dx$ .

EXAMPLE 2

Find the exact value of $\int_1^e x^2 \ln x \, dx$ .

1
We have a product of an algebraic function ( $x^2$ ) and a logarithmic function ( $\ln x$ ). We use integration by parts.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAIntegration

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Key terms for this topic — skim now; the Check step will test them.

Indefinite integral: The family of functions whose derivative is the integrand. It is written as $\int f(x) dx = F(x) + C$ , where $F'(x) = f(x)$ and $C$ is the constant of integration.
integral of $e^{ax+b}$: $\int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + C$ . A common mistake is forgetting the $\frac{1}{a}$ factor.
integration by parts: $\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$ . It is used to integrate the product of two functions.
integral of the form: This is a standard pattern that integrates to $\ln|f(x)| + C$ . Recognising this saves you from using substitution.

09Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 6 answered

10Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

11Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

close the tab only after these stick

The factor of $\frac{1}{a}$ appears when integrating a function of a linear argument $(ax+b)$ . This comes from reversing the chain rule.
Always include the constant of integration, $+C$ , for indefinite integrals.
The modulus sign in $\ln|ax+b|$ is crucial as the logarithm is only defined for positive inputs.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Question 1 of 29709/22 · Q5(b)

MathematicsMay/June 2024

(b)

Find the value of ∫₀² p(x)/(3x+2) dx, giving your answer in the form a+lnb where a and b are integers.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9709/22/M/J/24

Question 2 of 29709/31 · Q8

MathematicsOctober/November 2025

The variables x and y satisfy the differential equation (x²+1) dy/dx = kxe^(2y), where k is a constant. It is given that y = 0 when x = 0 and that y = -½ when x = 1. Solve the differential equation and find the exact value of y when x = √3.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9709/31/O/N/25

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

What's the difference between definite and indefinite integrals?

I always forget '+ C'. How important is it?

When do I use substitution versus integration by parts?

Why do the limits change when using substitution?

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do 9709/22 · Q5(b) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS3.4 Differentiation NEXT →3.6 Numerical solution of equations

∫

Discuss Integration

Ask, share and discuss with other Mathematics students

1June 2026 thresholds are live — post your raw marks and we will work out your gap2 #-1Result1 #1Results for AS0 #1One mark off a grade. Is a remark actually worth it?1 #

Create a post Visit s/9709

Revise Integration for Cambridge 9709 (3.5)

Looking for free Mathematics 9709 notes on Integration? This A-Level lesson covers official syllabus point 3.5 on Pure Mathematics 3. Use the visual steps, key terms, and quick checks below — then practise a real 9709 past-paper question with mark-scheme marking. Built for students searching "Integration 9709 revision", "9709 3.5 notes", or a free alternative to paid note sites.

Full Mathematics 9709 course Mark a past paper →9709 subject hub

3.5Mathematics22 min readPREMIUM

Integration

BY THE END, YOU CAN…

exam checklist — tick these off

1Integrate standard functions including exponential, logarithmic, and trigonometric forms.
2Apply integration by substitution and integration by parts to solve more complex integrals.
3Use definite integration to calculate the exact area under a curve or between two curves.
4Calculate the volume of a solid of revolution by rotating a region about the x-axis or y-axis.

STUDY MODE

READING

TYPEFACE

Literata — a reading serif made for e-books; set like a textbook, sits well beside maths.

SIZE

AIRY SPACING

Wider letter and line spacing. The one change with solid evidence for readers who lose their place — worth a try whoever you are.

Pick whatever reads fastest for you — it differs from person to person, by up to a third.

EXPLAIN SIMPLER

The same notes in plain English — no jargon. Key points and exam tips stay.

DIAGRAM NOTES

A small sketch beside each block of notes.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Un-doing Derivatives to Find Area

THINK OF IT LIKE…

1
Integration reverses differentiation — find the antiderivative F(x) with F′(x) = f(x).
2
A definite integral ∫ₐᵇ f(x) dx equals the signed area under the curve.
3
Evaluate F(b) − F(a) after finding F(x); always substitute limits explicitly.
4
Integration by parts: ∫ u dv = uv − ∫ v du — choose u using LIATE.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

1 more simulation for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

1 simulation

GeoGebraCore9709 3.5
Area between two curves
Drag the red points on the x-axis to set the limits, or type in your own pair of functions.
Try this
1. Drag the two red points to the x-values where the curves cross.
2. Read the shaded area, then integrate (upper − lower) by hand between the same limits.
3. Push one limit past a crossing point and watch the running total start to fall.
Look for Area = ∫(upper − lower) dx between the intersection points; where the curves swap over, the integral has to be split.
Open on GeoGebra
Ben Roth · GeoGebra · GeoGebra Terms of Service

Key formulas

Tap any symbol to reveal exactly what it means and its units.

KEY RELATION

Tap a symbol — great for exam definitions

KEY RELATION

$\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Standard Integrals and Key Forms

The factor of $\frac{1}{a}$ appears when integrating a function of a linear argument $(ax+b)$ . This comes from reversing the chain rule.
Always include the constant of integration, $+C$ , for indefinite integrals.
The modulus sign in $\ln|ax+b|$ is crucial as the logarithm is only defined for positive inputs.

Not clear?

Integration by Substitution

Not clear?

Integration by Parts

The Integration by Parts Formula: \ $\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$

Not clear?

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

reveal slowly — mark each step

EXAMPLE 1

Find the exact area of the region enclosed by the curve $y = x\sqrt{x^2+5}$ , the x-axis, and the lines $x=0$ and $x=2$ .

1
The area is given by the definite integral $A = \int_0^2 x\sqrt{x^2+5} \, dx$ .

EXAMPLE 2

Find the exact value of $\int_1^e x^2 \ln x \, dx$ .

1
We have a product of an algebraic function ( $x^2$ ) and a logarithmic function ( $\ln x$ ). We use integration by parts.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAIntegration

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Key terms for this topic — skim now; the Check step will test them.

Indefinite integral: The family of functions whose derivative is the integrand. It is written as $\int f(x) dx = F(x) + C$ , where $F'(x) = f(x)$ and $C$ is the constant of integration.
integral of $e^{ax+b}$: $\int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + C$ . A common mistake is forgetting the $\frac{1}{a}$ factor.
integration by parts: $\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$ . It is used to integrate the product of two functions.
integral of the form: This is a standard pattern that integrates to $\ln|f(x)| + C$ . Recognising this saves you from using substitution.

09Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 6 answered

10Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

11Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

close the tab only after these stick

The factor of $\frac{1}{a}$ appears when integrating a function of a linear argument $(ax+b)$ . This comes from reversing the chain rule.
Always include the constant of integration, $+C$ , for indefinite integrals.
The modulus sign in $\ln|ax+b|$ is crucial as the logarithm is only defined for positive inputs.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Question 1 of 29709/22 · Q5(b)

MathematicsMay/June 2024

(b)

Find the value of ∫₀² p(x)/(3x+2) dx, giving your answer in the form a+lnb where a and b are integers.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9709/22/M/J/24

Question 2 of 29709/31 · Q8

MathematicsOctober/November 2025

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9709/31/O/N/25

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

What's the difference between definite and indefinite integrals?

I always forget '+ C'. How important is it?

When do I use substitution versus integration by parts?

Why do the limits change when using substitution?

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do 9709/22 · Q5(b) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS3.4 Differentiation NEXT →3.6 Numerical solution of equations

∫

Discuss Integration

Ask, share and discuss with other Mathematics students

1June 2026 thresholds are live — post your raw marks and we will work out your gap2 #-1Result1 #1Results for AS0 #1One mark off a grade. Is a remark actually worth it?1 #

Create a post Visit s/9709

Revise Integration for Cambridge 9709 (3.5)

In simple terms

Explore the concept

Simulations

Area between two curves

Key formulas

Full topic notes

Standard Integrals and Key Forms

Integration by Substitution

Integration by Parts

Worked examples

How it all connects

Glossary

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Integration

Revise Integration for Cambridge 9709 (3.5)

In simple terms

Explore the concept

Simulations

Area between two curves

Key formulas

Full topic notes

Standard Integrals and Key Forms

Integration by Substitution

Integration by Parts

Worked examples

How it all connects

Glossary

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Integration