Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can explain why some integrands must be rewritten in an equivalent form before any rule applies. (Guide: “Why rewrite before integrating?”)
- I can decide from the degrees of top and bottom whether long division comes first. (Guide: decision table; Practice Q1, Q7(a))
- I can use the discriminant to decide whether a quadratic leads to an arctan form, and explain why x² − 4 does not. (Guide: “When does arctan apply?”; Practice Q2, Q7(d))
- I can explain where the factor 1/a in the arctan form comes from. (Guide: “Where the 1/a comes from”; Practice Q3)
Skills
- I can divide one polynomial by another, write quotient + remainder/divisor and check by multiplying back. (Guide: “Tool 1: long division”; Practice Q1, Q5(a))
- I can integrate the result of a division, including a remainder term that gives ln. (Guide: Worked example 1; Practice Q5(b), Q7(b))
- I can complete the square, including when the x² term is negative. (Guide: “Tool 2: completing the square”; Practice Q2, Q4, Q6(b))
- I can use the arctan and arcsin forms with the correct a, and change limits in a definite integral. (Guide: Worked example 2; Practice Q3, Q4)
- I can split a linear numerator into k × (derivative of the bottom) + constant and integrate both parts. (Guide: Worked example 3; Practice Q6)
Reasoning
- I can check an antiderivative by differentiating it. (Guide: Worked example 3)
- I can check a definite integral with a quick size estimate or a bounding function. (Guide: Worked examples 1 and 2; Practice Q7(c))
- I can explain why ∫ p(x)/q(x) dx is not (∫ p dx)/(∫ q dx). (Guide: “Common misconceptions”; Practice Q1)
All ticked? Move on to Topic 6.14, Selecting Techniques for Antidifferentiation, or return to the Calculus AB roadmap or the Calculus BC roadmap.