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IB Maths AI SL · Unit 5: Calculus

IB Maths AI SL Integration Questions

Exam-style IB Maths AI SL integration questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.

Practise Integration questions → AI SL formula booklet

What you need to know

SL AI keeps integration applied — 'find the area between the curve and the x-axis'. Watch for signed area (below the x-axis counts negative) — a classic trap. Integration and area under a curve overview →

What's examined in AI SL integration

The question bank covers these integration question types (number of questions in brackets):

Key formulas

Integral of x^n
\(\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C, \quad n \ne -1\)
Trapezoidal rule (approx. area)
\(A \approx \tfrac{h}{2}\bigl(y_0 + y_n + 2(y_1 + y_2 + \cdots + y_{n-1})\bigr),\ h = \dfrac{b-a}{n}\)
Area between curve and x-axis
\(A = \int_a^b y\, dx\)

In the same notation as the IB formula booklet. All AI SL formulas →

Integration worked examples

Worked example 1: Indefinite integration · easy

Find the exact indefinite integral $\int (6x^2 - 4x + 3)\,dx$.

Solution

1. Reverse power rule: add 1 to each exponent, divide by the new exponent.

2. Term-by-term: $\tfrac{6x^3}{3} - \tfrac{4x^2}{2} + \tfrac{3x^1}{1}$.

3. Simplify: $2x^3 - 2x^2 + 3x$.

4. Add $C$.

5. State: $\mathbf{2x^3 - 2x^2 + 3x + C}$.

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

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

Solution

1. Integrate: $y = \int (3x^2 - 8x)\,dx = x^3 - 4x^2 + C$.

2. Substitute $(2,5)$: $5 = 8 - 16 + C$.

3. Simplify: $5 = -8 + C$.

4. Solve: $C = 13$.

5. State: $\mathbf{y = x^3 - 4x^2 + 13}$.

Examiner tip: Given $\tfrac{dy}{dx}$ and a boundary point, always integrate first, then use the point to pin down $C$.

Worked example 3: Definite integral for area · hard

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

Solution

1. Roots: $-x^2 + 6x - 5 = 0 \Rightarrow x^2 - 6x + 5 = 0 \Rightarrow (x-1)(x-5)=0$.

2. Intercepts: $x = 1$ and $x = 5$.

3. Set up: Area $= \int_1^5 (-x^2 + 6x - 5)\,dx$.

4. GDC: use the definite integral tool.

5. Evaluate: $\mathbf{\tfrac{32}{3}\text{ units}^2}$ ($\approx 10.7$).

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

Try these IB Maths AI SL integration questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.

Question 1 · medium · 6 marks · Paper 2

The curve \(y = -x^2 + 6x - 5\) forms an enclosed region with the \(x\)-axis.

  1. Use your graphic display calculator to find the \(x\)-intercepts (roots) of the curve.

  2. Write down the definite integral that represents the area of the enclosed region.

  3. Evaluate the area of this region.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

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

  1. Determine the times at which the particle is momentarily at rest.

  2. Use your graphic display calculator to find the total distance travelled by the particle in the first \(6\) seconds.

Attempt it and see the mark scheme →

All 28 integration questions with mark schemes →

FAQ

How many IB Maths AI SL integration questions are there?

There are 28 exam-style integration questions in the AI SL question bank (Paper 1: 16 · Paper 2: 12), graded 2 easy, 12 medium, 6 hard, 4 very hard, 4 starter. 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: 16 · Paper 2: 12. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

More AI SL Unit 5 topics

Integration in other IB Maths courses

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