IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus
Advanced Integration: Substitution and Integration by Parts
Choose between substitution and integration by parts, including repeated/cyclic applications.
Learning goal
Use HL substitution and integration by parts, including repeated integration by parts.
Syllabus link
AA HL: AHL 5.16 · Topic 5 Calculus.
Big idea
Method selection comes from structure: compositions suggest substitution; products with a differentiable simplification often suggest integration by parts.
Key relationship
\(\int u\,dv=uv-\int v\,du\)
Core idea
Method selection comes from structure: compositions suggest substitution; products with a differentiable simplification often suggest integration by parts.
Use HL substitution and integration by parts, including repeated integration by parts.
Key results and method
- For substitution, transform the full integral to the new variable.
- For integration by parts, choose \(u\) and \(dv\), then calculate \(du\) and \(v\).
- For repeated/cyclic products, keep the algebra organised and solve for the original integral if it reappears.
Worked example
Evaluate \(\int x e^x dx\).
- Choose \(u=x\), \(dv=e^x dx\).
- Then \(du=dx\), \(v=e^x\).
- Apply \(uv-\int vdu\).
Practice
Try these questions before moving on.
- Integrate \(x\sin x\).
- Integrate \(\ln x\) by viewing it as \(1\cdot\ln x\).
- Use repeated integration by parts for \(x^2e^x\).
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