IB Maths AA SL · Unit 5: Calculus
IB Maths AA SL Integration Questions
Exam-style IB Maths AA 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.
- 21 questions
- Paper 1: 14
- Paper 2: 7
- 5 easy
- 7 medium
- 5 hard
- 3 very hard
- 1 starter
- 3 worked examples
Practise Integration questions →
AA SL formula booklet
What's examined in AA SL integration
The question bank covers these integration question types (number of questions in brackets):
- Basic Indefinite Integrals (10)
- Integration by Substitution (8)
- Definite Integrals & Area (3)
Key formulas
- Integral of x^n
- \(\int x^n\, dx = \dfrac{x^{n+1}}{n+1} + C,\ n \ne -1\)
- Standard integrals
- \(\int \sin x\, dx = -\cos x + C,\ \int \cos x\, dx = \sin x + C,\ \int e^x\, dx = e^x + C,\ \int \dfrac{1}{x}\, dx = \ln|x| + C\)
- Area between a curve and the x-axis
- \(A = \int_a^b y\, dx\)
- Volume of revolution (about x-axis)
- \(V = \pi \int_a^b y^2\, dx\)
In the same notation as the IB formula booklet. All AA SL formulas →
Integration worked examples
Worked example 1: Polynomial and exponential integration · easy
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: $\mathbf{2x^3 - 2x^2 + e^x + C}$.
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 $\int (3x - 1)^3 \,dx$.
1. Recognise the form $(ax+b)^n$.
2. Recall the rule: $\int (ax+b)^n \,dx = \frac{1}{a} \cdot \frac{(ax+b)^{n+1}}{n+1} + C$.
3. Identify $a = 3$, $b = -1$, $n = 3$.
4. Apply the rule: add $1$ to the power (now $4$) and divide by the new power.
5. Divide by the derivative of the inner linear function ($3$): $\frac{1}{3} \times \frac{(3x - 1)^4}{4}$.
6. Simplify and add $C$: $\mathbf{\frac{1}{12}(3x - 1)^4 + C}$.
Examiner tip: The $\frac{1}{a}$ adjustment ONLY works when the inner function is strictly linear (highest power of $x$ is $1$). 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 $\int x e^{x^2 - 2} \,dx$.
1. Identify the pattern $\int k \cdot g'(x) f(g(x)) \,dx$, with the outer an exponential.
2. Set the inner function: $g(x) = x^2 - 2$.
3. Differentiate: $g'(x) = 2x$.
4. Compare: the integrand has $x$, which is $\frac{1}{2}$ of $g'(x) = 2x$. So the adjustment factor is $\frac{1}{2}$.
5. Construct the anti-derivative: $\frac{1}{2} e^{g(x)}$.
6. State: $\mathbf{\frac{1}{2} e^{x^2 - 2} + C}$.
Examiner tip: Always sanity-check integration by differentiating: $\frac{d}{dx}\!\left[\tfrac{1}{2}e^{x^2-2}\right] = \tfrac{1}{2} \cdot 2x \cdot e^{x^2-2} = xe^{x^2-2}$ ✓.
Try these IB Maths AA 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 · easy · 5 marks · Paper 2
The gradient function of a curve is given by \(\frac{dy}{dx} = 4x^3 - 2x\). 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.
Question 2 · medium · 5 marks · Paper 1
Given that \(f(x) = 2x^3 + 4x\), find \(f'(x)\). [5]
Hence, or otherwise, find the indefinite integral:
\[\int \frac{3x^2 + 2}{2x^3 + 4x} \,dx\]
Question 3 · hard · 5 marks · Paper 1
Find an expression for \(y\) given that: [9]
\[\frac{dy}{dx} = x e^{x^2 - 2}\]
and the curve passes through the point \((-\sqrt{2}, 3)\).
All 21 integration questions with mark schemes →
FAQ
How many IB Maths AA SL integration questions are there?
There are 21 exam-style integration questions in the AA SL question bank (Paper 1: 14 · Paper 2: 7), graded 5 easy, 7 medium, 5 hard, 3 very hard, 1 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: 14 · Paper 2: 7. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA 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.
Integration in other IB Maths courses
← All IB Maths AA SL topics