Integration

Year 1 / ASYear 2 / A-Level All Boards (AQA, Edexcel, OCR, WJEC, CCEA) AQA

A-Level Mathematics revision: Integration. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.

Exam Board:

📌 Key Points

Key Fact: Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n != -1)

Key Fact: Special: ∫1/x dx = ln|x| + C, ∫eˣ dx = eˣ + C

Key Fact: Trig: ∫sin x dx = -cos x + C, ∫cos x dx = sin x + C, ∫sec^2x dx = tan x + C

Key Fact: Reverse chain: ∫f'(x)[f(x)]ⁿ dx = [f(x)]ⁿ⁺¹/(n+1) + C

Key Fact: Substitution: let u = f(x), dx = du/f'(x), change limits if definite

Key Fact: By parts: ∫u dv = uv - ∫v du (LIATE for choosing u)

Key Fact: Definite: ∫ₐᵇ f(x)dx = F(b) - F(a), area = ∫|f(x)|dx

Key Fact: Area between curves: ∫(top - bottom)dx over intersection interval

Key Fact: Volume of revolution: V = pi∫y^2 dx (about x-axis) or pi∫x^2 dy (about y-axis)

🎯 Learning Objectives

Integrate polynomials, rational and negative powers
Use reverse chain rule for composite functions
Integrate trigonometric, exponential and logarithmic functions
Apply substitution and integration by parts
Evaluate definite integrals and find area under curves
Find area between two curves
Calculate volumes of revolution (disc/washer method)
Solve differential equations by separation of variables

💡 Worked Example

Exam-Style Question

Question: Evaluate ∫₀^2 xsqrt(4 - x^2) dx using substitution u = 4 - x^2

Model Answer:

u = 4 - x^2, du = -2x dx -> x dx = -½ du. Limits: x=0->u=4, x=2->u=0. ∫₄⁰ sqrtu (-½)du = ½∫₀⁴ u^½ du = ½[⅔u^⅔]₀⁴ = ⅓(8) = 8/3

❓ Practice Questions

Model answers are being added progressively - questions marked ✗ don't have one yet. Cross-check with your teacher or the official mark scheme.

Questions:

∫(3x^2 - 2x + 5)e^{x^3-x^2+5x} dx✗ answer coming soon
∫x cos x dx (by parts)✗ answer coming soon
∫₀¹ x/(1+x^2)^2 dx✗ answer coming soon
Area between y = x^2 and y = 2x - x^2✗ answer coming soon
Volume when y = sqrtx, 0<=x<=4 rotated about x-axis✗ answer coming soon
Solve dy/dx = 2xy, y(0)=3✗ answer coming soon

🎬 Video Resources

Integration - Overview -- YouTube Search
Integration - Exam Questions -- YouTube Search

📄 Past Papers & Exam Resources

AQA A-Level Mathematics - Integration -- AQA
Edexcel A-Level Mathematics - Integration -- Edexcel
OCR A-Level Mathematics - Integration -- OCR

🔗 Further Reading & Resources

Physics & Maths Tutor - Integration -- Free notes, worksheets, and past paper questions by topic
Dr Frost Maths - Integration -- Free resources, slides, and interactive questions
BBC Bitesize - Integration -- BBC Bitesize revision resources

📚 Lesson Plan (50 minutes)

Starter (5 min): Recall prior knowledge of integration with quick questions.
Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
Key points review (5 min): Revisit the key points together, confirming understanding.
Worked example (10 min): Model the example question: Evaluate ∫₀^2 xsqrt(4 - x^2) dx using substitution u = 4 - x^2. Solution: u = 4 - x^2, du = -2x dx -> x dx = -½ du. Limits: x=0->u=4, x=2->u=0. ∫₄⁰ sqrtu (-½)du = ½∫₀⁴ u^½ du = ½[⅔u^⅔]₀⁴ = ⅓(8) = 8/3
Practice (10 min): Students attempt the practice questions independently; circulate and support.
Plenary (5 min): Review answers and address misconceptions.

🏠 Homework

∫(3x^2 - 2x + 5)e^{x^3-x^2+5x} dx
∫x cos x dx (by parts)
∫₀¹ x/(1+x^2)^2 dx
Area between y = x^2 and y = 2x - x^2
Volume when y = sqrtx, 0<=x<=4 rotated about x-axis
Solve dy/dx = 2xy, y(0)=3

🧾 Assessment

Check practice answers against the model answer; use the built-in practice questions as formative assessment.

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