The velocity of a particle, $v$ in $\text{m s}^{-1}$ , is given by $v(t) = t^2 - 7t + 10$ for $t \ge 0$ . Determine the times at which the particle is momentarily at rest. Use your graphic display calculator to find the total distance travelled by the particle in the first $6$ seconds.

Particle at rest when $v(t) = 0$ . (M1) Factorise: $t^2 - 7t + 10 = 0 \implies (t-2)(t-5) = 0$ . $t = 2\text{ s}$ and $t = 5\text{ s}$ . A1A1 Velocity is negative between $t=2$ and $t=5$ , so integrate the absolute value of velocity: Distance $= \int_{0}^{6} |t^2 - 7t + 10| \, dt$ (CG50 MENU 1 : OPTN -> NUMERIC -> Abs inside the integral). (M1) Check by intervals: $\frac{26}{3} + \frac{9}{2} + \frac{11}{6} = 15$ . Total distance $= 15.0\text{ m}$ . A1

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IB Maths AI SL · Unit 5: Calculus

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

Updated October 2026 · By Pete Bromfield

Integration reverses differentiation: $\int a x^{n} d x = \frac{a}{n + 1} x^{n + 1} + C$ for $n \neq - 1$ . In AI SL you integrate polynomials, find $C$ from a known point, find areas under curves with definite integrals on your GDC, and estimate areas with the trapezoidal rule.

35 questions
Paper 1: 20
Paper 2: 15
3 worked examples

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Notes
Formulas
Worked examples
Questions
FAQ

Integration notes

Indefinite integrals

Add one to the power and divide by the new power: $\int (6 x^{2} + 4) d x = 2 x^{3} + 4 x + C$ . Never leave out $+ C$ . Check by differentiating your answer.

Finding the constant of integration

A known point on the curve fixes $C$ . If $f^{'} (x) = 6 x^{2} + 4$ and the curve passes through $(1, 10)$ , then $2 + 4 + C = 10$ , so $C = 4$ and $f (x) = 2 x^{3} + 4 x + 4$ .

Area under a curve

The area between $y = f (x)$ , the $x$ -axis and $x = a$ to $x = b$ (with $f (x) \geq 0$ ) is $\int_{a}^{b} f (x) d x$ . Evaluate it on your GDC and write the integral down first. For example $\int_{0}^{2} (x^{2} + 1) d x = \frac{14}{3} \approx 4.67$ .

The trapezoidal rule

With $n$ strips of width $h = \frac{b - a}{n}$ : area $\approx \frac{1}{2} h [(y_{0} + y_{n}) + 2 (y_{1} + \dots + y_{n - 1})]$ . Count strips, not $x$ -values: 4 strips need 5 values. For $y = x^{2} + 1$ from 0 to 2 with 4 strips, the estimate is $4.75$ , a little above the exact $4.67$ because the curve bends upwards.

Key formulas

Integral of $x^{n}$ , $n \neq - 1$ In the booklet: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$
Area, $y \geq 0$ In the booklet: $A = \int_{a}^{b} y d x$
Trapezoidal rule, $h = \frac{b - a}{n}$ In the booklet: $\int_{a}^{b} y d x \approx \frac{1}{2} h [(y_{0} + y_{n}) + 2 (y_{1} + \dots + y_{n - 1})]$
Boundary condition: find $C$ by substituting a known point into $\int f^{'} (x) d x$ Learn it

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 AI SL formulas →

Integration worked examples

Worked example 1: Indefinite integration · easy

Find the exact indefinite integral $\int (6 x^{2} - 4 x + 3) d x$ .

Solution

5 steps

Examiner tip: Always append "

+ C

" on an indefinite integral. Forgetting it costs a mark every time.

Worked example 2: Finding the constant of integration · medium

$\frac{d y}{d x} = 3 x^{2} - 8 x$ and the curve passes through $(2, 5)$ . Find the exact equation of the curve.

Solution

5 steps · 1 to fill in

Examiner tip: Given

\frac{d y}{d x}

and a boundary point, always integrate first, then use the point to pin down

C

Worked example 3: Definite integral for area · hard

$y = - x^{2} + 6 x - 5$ encloses a region above the $x$ -axis. Find the $x$ -intercepts and the exact area of the region.

Solution

5 steps · 1 to fill in

Examiner tip: In AI SL, once you have the correct definite integral written down, evaluate it on the GDC to avoid arithmetic slips.

Integration exam-style questions

3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Each has a full mark scheme in the practice area.

Question 1 · easy · 3 marks · Paper 1

Use the trapezoidal rule with $4$ intervals to estimate $\int_{0}^{2} (x^{2} + 1) d x$ .

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

The velocity of a particle, $v$ in ${m s}^{- 1}$ , is given by $v (t) = t^{2} - 7 t + 10$ for $t \geq 0$ .

Determine the times at which the particle is momentarily at rest.
Use your graphic display calculator to find the total distance travelled by the particle in the first $6$ seconds.

Full worked solution Free, no sign-in

Particle at rest when $v (t) = 0$ . (M1)
Factorise: $t^{2} - 7 t + 10 = 0 ⟹ (t - 2) (t - 5) = 0$ .
$t = 2 s$ and $t = 5 s$ . A1A1
Velocity is negative between $t = 2$ and $t = 5$ , so integrate the absolute value of velocity:
Distance $= \int_{0}^{6} | t^{2} - 7 t + 10 | d t$ (CG50 MENU 1: OPTN -> NUMERIC -> Abs inside the integral). (M1)
Check by intervals: $\frac{26}{3} + \frac{9}{2} + \frac{11}{6} = 15$ .
Total distance $= 15.0 m$ . A1

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

Question 3 · hard · 5 marks · Paper 1

Find the exact value of the constant $k$ ( $k > 0$ ) such that: $\int_{0}^{k} 3 x^{2} d x = 64$

Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 5 pages · 229 KB

All 35 integration questions →

Where students lose marks on integration

Examiner's noteTrapezoidal rule and integrals: heights, strip width and dx

Where marks are lost: Candidates substituted x-values instead of heights, used the wrong strip width (for example the number of values instead of the number of strips), misread the booklet formula, substituted limits when an indefinite integral was asked, omitted dx, or used limits outside the domain. Exact answers were rarely given when asked.

Examiners flagged this in 3 exam series (May 2024, November 2024, 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 every AI paper. 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 trapezoidal rule in the IB formula booklet?

Yes, the AI SL booklet gives the trapezoidal rule and the integral of xⁿ. Remember that n is the number of strips, so you need n + 1 y-values.

Can I use my GDC for definite integrals in AI SL?

Yes: a GDC is allowed on every AI paper, and definite integrals and areas are normally found with it. Write down the integral you entered, with its limits, before the answer.

How many IB Maths AI SL integration questions are there?

There are 35 exam-style integration questions in the AI SL question bank (Paper 1: 20 · Paper 2: 15), graded 4 easy, 19 medium, 8 hard, 4 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: 20 · Paper 2: 15. Practise with your GDC — AI papers expect calculator methods throughout.

Next steps

More AI SL Unit 5: Differentiation · Limits, Derivatives & Rates of Change · Tangents & Normals

Integration in other courses: AA SL

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Definite integrals & area under a curve.

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

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← All IB Maths AI SL topics

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

Try it first

Integration notes

Indefinite integrals

Finding the constant of integration

Area under a curve

The trapezoidal rule

Key formulas

Integration worked examples

Worked example 1: Indefinite integration · easy

Solution

Worked example 2: Finding the constant of integration · medium

Solution

Worked example 3: Definite integral for area · hard

Solution

Integration exam-style questions

Question 1 · easy · 3 marks · Paper 1

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