BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (Topics 2.8, 6.7, 6.8, 6.9) are on the Calculus BC roadmap.
Understanding
- I can derive ∫ u dv = uv − ∫ v du from the product rule, step by step. (Guide: “Deriving the formula from the product rule”)
- I can explain why integration by parts trades one integral for another, and when the trade is worth it. (Guide: “Choosing u and dv”; Practice Q3)
- I can recognise when substitution, not parts, is the right method. (Guide: “Why we need a new technique”; Common misconceptions)
Skills
- I can find an indefinite integral such as ∫ x e^(2x) dx by parts, including the chain-rule factor in v. (Guide: Worked example 1; Practice Q1)
- I can integrate ln x by taking dv = dx. (Guide: “When dv = dx”; Practice Q2)
- I can apply parts twice to integrals such as ∫ x² sin x dx. (Guide: “Repeated parts”; Practice Q5)
- I can solve for an integral that returns to itself, such as ∫ eˣ cos x dx. (Guide: “Integrals that come back”; Practice Q4)
- I can evaluate a definite integral by parts, applying the limits to both uv and the new integral. (Guide: Worked example 2; Practice Q2, Q6)
- I can check any antiderivative by differentiating it with the product rule. (Practice Q5, Q6(d))
Reasoning
- I can justify a choice of u and dv using the two conditions (dv integrable, du simpler), and explain a case where LIATE fails. (Guide: “Choosing u and dv”; Practice Q3)
- I can set up and evaluate an accumulation problem with a product integrand, and state the answer with units. (Guide: Worked example 2; Practice Q7)
- I can use a definite integral of a rate to justify a claim about a total amount, such as energy delivered. (Practice Q7(c))
All ticked? Move on to Topic 6.12, Integrating Using Linear Partial Fractions, or return to the Calculus BC roadmap.