What's the difference between definite and indefinite integrals?

An indefinite integral, $\int f(x) dx$, gives a function (or family of functions, $F(x)+C$) representing the antiderivative. A definite integral, $\int_a^b f(x) dx$, gives a single numerical value representing the signed area under the curve between two points.

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

For indefinite integrals, it is critically important and you will lose a mark if you forget it. It represents an entire family of functions that have the same derivative. For definite integrals, the '+ C' cancels out when you evaluate $F(b) - F(a)$, so it's not needed in the final calculation, but the underlying indefinite integral technically still has it.

When do I use substitution versus integration by parts?

Use substitution when the integrand contains a function and its derivative (or something close to it), often in a 'function-of-a-function' or composite form. Use integration by parts when the integrand is a product of two unrelated functions, like $x e^x$ or $x^2 \sin x$.

Why do the limits change when using substitution?

The original limits are $x$-values. When you change the entire integral to be in terms of a new variable, $u$, the limits must also be expressed in terms of $u$. This allows you to complete the calculation without ever having to substitute back to $x$, which is more efficient and less error-prone.

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?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Integration.

Integration 9709 — Free A-Level Maths

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…

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.

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.

Key formulas

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

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

Tap a symbol — great for exam definitions

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

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.

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.

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

Worked examples

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

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

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is an indefinite integral?

TAP OR SPACE TO FLIP

ANSWER

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.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
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.

Real past paper[0]

Practice Integration Problems

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

Extra simulations & links

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

Frequently asked

✓

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 Practice Integration Problems 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

Integration

BY THE END, YOU CAN…

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.

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.

Key formulas

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

Tap a symbol — great for exam definitions

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

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.

Integration by Substitution

Integration by Parts

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

Worked examples

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

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

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is an indefinite integral?

TAP OR SPACE TO FLIP

ANSWER

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.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
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.

Real past paper[0]

Practice Integration Problems

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

Extra simulations & links

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

Frequently asked

✓

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 Practice Integration Problems 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

Revise Integration for Cambridge 9709 (3.5)

In simple terms

Explore the concept

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

Revision flashcards

Key takeaways

Practice — then mark it

Practice Integration Problems

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

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

Revision flashcards

Key takeaways

Practice — then mark it

Practice Integration Problems

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Integration