Find the indefinite integral $\int \left( 6x^2 - 4x + e^x \right) \,dx$.

1. Integrate the first term (power rule): $\frac{6x^3}{3} = 2x^3$. 2. Integrate the second term (power rule): $\frac{-4x^2}{2} = -2x^2$. 3. Recall $\int e^x \,dx = e^x$. 4. Combine : $2x^3 - 2x^2 + e^x$. 5. Add the constant of integration $C$ — required for indefinite integrals. 6. State : $\mathbf{2x^3 - 2x^2 + e^x + C}$.

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IB Maths AA SL · Unit 5: Calculus

Integration (IB Maths AA SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

Integration reverses differentiation: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ for $n \neq - 1$ , plus the standard integrals of $\frac{1}{x}$ , $e^{x}$ , $\sin x$ and $\cos x$ . Every indefinite integral needs $+ C$ . In AA SL you also integrate a linear inside such as $e^{3 x + 1}$ , use the reverse chain rule by inspection, and find $C$ from a known point.

41 questions
Paper 1: 30
Paper 2: 11
3 worked examples

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Notes
Formulas
Worked examples
Questions
FAQ

Integration notes

Antiderivatives and the power rule

Add one to the power and divide by the new power: $\int (6 x^{2} - 4 x) d x = 2 x^{3} - 2 x^{2} + C$ . Rewrite roots and fractions as powers first: $\int \frac{1}{\sqrt{x}} d x = \int x^{- 1 / 2} d x = 2 \sqrt{x} + C$ .

Standard integrals

$\int \frac{1}{x} d x = \ln | x | + C$ , $\int e^{x} d x = e^{x} + C$ , $\int \sin x d x = - \cos x + C$ and $\int \cos x d x = \sin x + C$ . These are in the formula booklet; watch the minus sign on $\sin x$ .

A linear function inside

For $f (a x + b)$ , integrate as usual and divide by $a$ : $\int e^{3 x + 1} d x = \frac{1}{3} e^{3 x + 1} + C$ and $\int \cos 2 x d x = \frac{1}{2} \sin 2 x + C$ . The divide-by- $a$ shortcut works only when the inside is linear.

The reverse chain rule (by inspection)

Look for a function and the derivative of its inside: $\int 2 x (x^{2} + 1)^{4} d x = \frac{1}{5} (x^{2} + 1)^{5} + C$ . Guess the answer, differentiate your guess, and adjust the constant until it matches.

Finding the constant of integration

A known point fixes $C$ . If $f^{'} (x) = 3 x^{2}$ and $f (1) = 5$ , then $f (x) = x^{3} + C$ and $1 + C = 5$ , so $f (x) = x^{3} + 4$ . For areas and definite integrals, see definite integrals.

Key formulas

Integral of $x^{n}$ , $n \neq - 1$ In the booklet: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$
Standard integrals In the booklet: $\int \frac{1}{x} d x = \ln | x | + C, \int \sin x d x = - \cos x + C, \int \cos x d x = \sin x + C, \int e^{x} d x = e^{x} + C$
Linear inside Learn it: $\int f (a x + b) d x = \frac{1}{a} F (a x + b) + C$
Reverse chain rule Learn it: $\int f^{'} (g (x)) g^{'} (x) d x = f (g (x)) + C$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Integration worked examples

Worked example 1: Polynomial and exponential integration · easy

Find the indefinite integral $\int (6 x^{2} - 4 x + e^{x}) d x$ .

Solution

6 steps

Examiner tip: Do not forget the

+ C

. Examiners typically penalise one mark for a missing constant of integration on an indefinite integral question.

Worked example 2: Reverse chain rule on a linear composite · medium

By using integration by inspection (the reverse chain rule) on linear composites, find the indefinite integral $\int (3 x - 1)^{3} d x$ .

Solution

6 steps · 1 to fill in

Examiner tip: The

\frac{1}{a}

adjustment ONLY works when the inner function is strictly linear (highest power of

x

1

). It does NOT work for something like

(x^{2} - 1)^{3}

Worked example 3: Reverse chain rule with a non-linear inner · hard

Use integration by inspection (the reverse chain rule) to find the indefinite integral $\int x e^{x^{2} - 2} d x$ .

Solution

6 steps

Examiner tip: Always sanity-check integration by differentiating:

\frac{d}{d x} [\frac{1}{2} e^{x^{2} - 2}] = \frac{1}{2} \cdot 2 x \cdot e^{x^{2} - 2} = x e^{x^{2} - 2}

✓.

Integration exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 5 marks · Paper 2

The gradient function of a curve is given by $\frac{d y}{d x} = 4 x^{3} - 2 x$ . The curve passes through the point $(2, 10)$ .

Find the general expression for $y$ by integrating the gradient function.
Use the given boundary condition to find the constant of integration, $C$ , and write down the exact equation of the curve.

Check your answer in practice →

Question 2 · medium · 5 marks · Paper 1

Given that $f (x) = 2 x^{3} + 4 x$ , find $f^{'} (x)$ . [5] Hence, or otherwise, find the indefinite integral: $\int \frac{3 x^{2} + 2}{2 x^{3} + 4 x} d x$

Full worked solution Free, no sign-in

Differentiate: $f^{'} (x) = 6 x^{2} + 4$ A1A1
The numerator is half of this derivative: $3 x^{2} + 2 = \frac{1}{2} (6 x^{2} + 4)$ (M1)
The integral has the form $\int \frac{k f^{'} (x)}{f (x)} d x = k \ln | f (x) | + C$ (M1)
$\int \frac{3 x^{2} + 2}{2 x^{3} + 4 x} d x = \frac{1}{2} \ln | 2 x^{3} + 4 x | + C$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 5 marks · Paper 1

Find an expression for $y$ given that: $\frac{d y}{d x} = x e^{x^{2} - 2}$ and the curve passes through the point $(- \sqrt{2}, 3)$ .

Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 199 KB

All 41 integration questions →

Where students lose marks on integration

Examiner's noteIntegration: missing +C, substituting into the derivative, and area limits

Where marks are lost: Given a derivative and a point, candidates substituted the point into the derivative, differentiated instead of integrating, or dropped the constant. Integrating e^(kx) without the 1/k, not recognising ∫ 1/x dx = ln x, not writing the final function, and substituting only one limit in an area were all reported.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Integration on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Numerical derivative and integral on your calculator

Integration FAQ

Is the integral of 1/x in the IB formula booklet?

Yes. The AA SL booklet lists the integrals of xⁿ, 1/x, sin x, cos x and eˣ. The rules for a linear inside and for integrating by inspection are not printed, so practise them.

Do I always need + C?

On every indefinite integral, yes: missing + C is one of the commonest lost marks. A definite integral (with limits) has no + C.

How many IB Maths AA SL integration questions are there?

There are 41 exam-style integration questions in the AA SL question bank (Paper 1: 30 · Paper 2: 11), graded 12 easy, 16 medium, 10 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is integration on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 30 · Paper 2: 11. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 5: Applications of Differentiation · Graphical Behaviour and the Second Derivative · Kinematics · Limits and Rates of Change · Differentiation Rules

Integration in other courses: AI SL

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Antiderivatives (integration) · Integration by substitution.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL calculus knowledge organiser

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Integration (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Integration notes

Antiderivatives and the power rule

Standard integrals

A linear function inside

The reverse chain rule (by inspection)

Finding the constant of integration

Key formulas

Integration worked examples

Worked example 1: Polynomial and exponential integration · easy

Solution

Worked example 2: Reverse chain rule on a linear composite · medium

Solution

Worked example 3: Reverse chain rule with a non-linear inner · hard

Solution

Integration exam-style questions

Question 1 · easy · 5 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 5 marks · Paper 1

Where students lose marks on integration

Integration on your calculator

Integration FAQ

Next steps

What else is free here