Free IB Maths AA SL lesson: Integration by substitution
A ready-to-teach IB Maths AA SL lesson on Integration by substitution (Unit 5 · Calculus): editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
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What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans6 questions, 22 marks: Fluency, Reasoning, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- Integration by substitution
Key ideas on the slides
- Choose u (often the inner function), find du/dx, and replace every x and dx with u and du.
- ∫ 2x(x² + 1)⁴ dx: with u = x² + 1, du = 2x dx, it becomes ∫ u⁴ du = ((x² + 1)⁵)/5 + C.
- For a definite integral, change the limits to u-values (then do not substitute back).
- Recognition ("inspection"): ∫ f′(x)[f(x)]ⁿ dx = ([f(x)]ⁿ⁺¹)/(n + 1) + C and ∫ f′(x)eᶠ⁽ˣ⁾ dx = eᶠ⁽ˣ⁾ + C.
By the end of the lesson, students should be able to answer a question like: Integrate 2x * cos(x^2) with respect to x using substitution.
Key vocabulary
- substitution: Replacing the variable to simplify an integral.
- limits: The start and end values, which change with the substitution.
- inspection: Recognising an integral as the reverse of a chain-rule derivative.
Common mistakes this lesson tackles
- Leaving some x terms after substituting, or forgetting to replace dx.
- Using the old x-limits with the new variable.
- Forgetting to substitute back in an indefinite integral.
The worked example
Find the indefinite integral ∫ (6x² − 4x + eˣ) dx.
- Integrate the first term (power rule): 6x³/3 = 2x³.
- Integrate the second term (power rule): (−4x²)/2 = −2x².
- Recall ∫ eˣ dx = eˣ.
- Combine: 2x³ − 2x² + eˣ.
- Add the constant of integration C — required for indefinite integrals.
- State: 2x³ − 2x² + eˣ + C.
Tip: Do not forget the + C. Examiners typically penalise one mark for a missing constant of integration on an indefinite integral question.
Calculator tips
- Check the value of the definite integral numerically.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Concavity & points of inflexion and Kinematics: displacement, velocity, acceleration |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (6 questions in 3 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens the IB Maths AA SL Unit 5: Integration practice questions.
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (with a School plan).
- This lesson's worksheet in three levels, with its answers, ready to print.
- Support, Core and Stretch versions and A/B copies of the worksheet from the full question bank, in the worksheet builder.
- Create lesson: this lesson rebuilt for your class's next scheme-of-work lesson, from the full bank.
School licences from €490 a year. Try everything free for 30 days as a teacher: no card.
Questions teachers ask
Is this lesson free?
The slides and the teacher notes are free to download with a free IB Math Revision teacher account (sign in with Google, Microsoft or email); no card. The printable worksheet in three levels and its answers are included with school plans and the free 30-day teacher trial.
Can I edit the slides?
Yes. The slides are an ordinary PowerPoint file: change, add or delete anything. They also open in Google Slides and Keynote. The logo and web address sit on the slide master, so your edits do not move them.
What do students use for homework?
The practice slide's QR code and web address take students to the IB Maths AA SL Unit 5: Integration practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.