Integration (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Integration (A2 Unit 3: Pure Mathematics B) explained
Integration techniques correspond directly to differentiation rules in reverse.
Read the full explanation
Integration by substitution reverses the chain rule: since d/dx[F(g(x))] = F'(g(x)) g'(x) = f(g(x)) g'(x), integrating both sides gives ∫ f(g(x)) g'(x) dx = F(g(x)) + c, which is executed by setting u = g(x). In particular, integrals of the shape ∫ [f'(x)/f(x)] dx integrate immediately to ln|f(x)| + c, and ∫ f'(x)[f(x)]ⁿ dx integrates to [f(x)]^(n+1) / (n + 1) + c. Integration by parts reverses the product rule: differentiating a product gives d/dx(uv) = u(dv/dx) + v(du/dx); integrating both sides yields uv = ∫ u(dv/dx) dx + ∫ v(du/dx) dx, which rearranges to the standard formula ∫ u(dv/dx) dx = uv − ∫ v(du/dx) dx.
Your focus
- Recognise and evaluate integrals of the form ∫ f'(x)/f(x) dx and ∫ f'(x)[f(x)]ⁿ dx by inspection.
- Explain the theoretical connection between integration by substitution and the chain rule.
- Explain the theoretical derivation of integration by parts from the product rule.
Integration (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- recognising and applying the reverse chain rule pattern ∫ [f'(x)/f(x)] dx = ln|f(x)| + c
- recognising and applying the pattern ∫ f'(x)[f(x)]ⁿ dx = [f(x)]^(n+1)/(n + 1) + c
- explaining integration by parts as the integration of the product rule for differentiation
Examiner Tips
- 💡Look for the derivative of the denominator in the numerator: ∫ [f'(x)/f(x)] dx is always ln|f(x)|.
- 💡Check if one factor in the integrand is proportional to the derivative of the inside of another factor.
Common Mistakes
- missing constant factors when recognizing reverse chain rule patterns (e.g. forgetting a factor of 1/2)
- attempting integration by parts on integrals that are straightforward reverse chain rule forms