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