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 substitution as the chain rule run backwards, using an example such as 3x² cos(x³). (Guide: “The idea: the chain rule run backwards”)
- I can spot a suitable inner function u because its derivative appears as a factor, up to a constant. (Guide: “How to spot a substitution”; Practice Q1, Q2)
- I can explain why a constant factor can be adjusted but a variable factor cannot. (Guide: “You cannot adjust a variable”; Practice Q5(c))
- I can explain why the limits change in a definite integral, using the two equal areas in Figure 1. (Guide: Figure 1; Practice Q6(d))
Skills
- I can find du, adjust for a constant and rewrite an integral entirely in u. (Guide: Worked example 1; Practice Q1, Q5(a))
- I can integrate a function of a linear expression ax + b by dividing by a. (Guide: “Linear inner functions”)
- I can replace a leftover x by solving u = g(x) for x. (Guide: Worked example 3; Practice Q4, Q7)
- I can evaluate a definite integral by changing the limits to u-values, including limits that come out in decreasing order. (Guide: Worked example 2; Practice Q3, Q6)
- I can write + C on every indefinite answer and check it by differentiating. (Guide: Worked example 1; Practice Q5(b))
Reasoning
- I can explain the error when x-limits are used with an antiderivative in u, and say what the right limits are. (Guide: “Common misconceptions”; Practice Q3, Q6(d))
- I can judge whether my answer is reasonable from the sign or size of the integrand. (Guide: Worked examples 2 and 3; Practice Q7(c))
- I can choose between changing the limits and returning to x, and use either route correctly. (Guide: “Two correct routes”; Practice Q6)
All ticked? Move on to Topic 6.10, Integrating Functions Using Long Division and Completing the Square, or return to the Calculus AB roadmap or the Calculus BC roadmap.