Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Integrate powers of x (indefinite integrals)
Calculus
CCSS HSF.BF.B.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Integrate powers of x (indefinite integrals).
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Differentiate axⁿ with the power rule
Leads to: Evaluate definite integrals; Find a curve from its gradient and a point
Materials: Graphing tool or graph paper, Scientific calculators, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Differentiate axⁿ with the power rule). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Differentiate y = −3x⁴. → -12*x^3Prompt: Multiply by the power, then reduce the power by 1.
- Mei differentiated 5x⁵ and wrote 25x⁶. What is the correct derivative? (25x⁶ / 20x⁴ / 5x⁴ / 25x⁴) → 25x⁴Prompt: The power goes down by one, not up.
- Differentiate y = −6x². → -12*x^1Prompt: Multiply by the power, then reduce the power by 1.
Hook · 5 min
- Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
- Share the objective: Integrate powers of x (indefinite integrals). Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find ∫ 20x³ dx. Leave out the + C.
- Step 1: 20x⁴/4
- Step 2: = 5x⁴ + C
- Answer: 5*x^4
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Find ∫ −8x dx. Leave out the + C. → -4*x^2Prompt: Add 1 to the power, then divide by the new power.
- Find ∫ (3x² − 8x − 6) dx. Leave out the + C. → x^3-4x^2-6xPrompt: Integrate each term separately.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Leo needs ∫ 4/x³ dx. Which answer is correct? (4·ln x + C / 4/x⁴ + C / −2/x² + C / 2/x² + C) → −2/x² + CPrompt: Rewrite as 4x^(−3).
- Find ∫ −14x⁶ dx. Leave out the + C. → -2*x^7Prompt: Add 1 to the power, then divide by the new power.
Watch out for
- Forgetting the constant of integration.
- Differentiating a product by multiplying the derivatives.
Support and stretch
Support
- Review first: Differentiate axⁿ with the power rule.
- Use concrete materials: graphing tool or graph paper.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Evaluate definite integrals; Find a curve from its gradient and a point.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder