Work through the list without notes. 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 name the signal that points to each technique: basic rule, rewrite, substitution, long division and completing the square. (Guide: “The toolkit and the signal for each tool”; Figure 1)
- I can explain why x/(x² + 9), 1/(x² + 9) and x²/(x² + 9) need three different techniques. (Guide: “Lookalike integrals”; Practice Q3)
- I can explain why some integrands, such as e^(−x²), have no antiderivative in familiar functions, and why their definite integrals still exist. (Guide: “When there is no formula”; Practice Q4)
Skills
- I can rewrite an integrand by splitting a fraction or writing roots as powers, then use the power rule. (Guide: Worked example 1(a); Practice Q5(a))
- I can spot an inner function with its derivative and integrate by substitution, adjusting for a constant factor. (Guide: Worked example 1(b); Practice Q1, Q5(b))
- I can use long division on a top-heavy rational function before integrating. (Guide: Worked example 1(c); Practice Q5(c), Q7(b))
- I can complete the square in a denominator with no real roots and integrate to an arctan. (Guide: Worked example 1(d); Practice Q2; arctan form also in Q6(b))
- I can split a numerator into “a multiple of the derivative” plus “a constant” and use two techniques in one integral. (Guide: Worked example 2)
- I can change the limits of a definite integral when I substitute. (Guide: Worked example 2; Practice Q6(a))
- I can use a calculator for a definite integral that has no closed-form antiderivative. (Practice Q6(c))
Reasoning
- I can justify my choice of technique in one sentence before I start. (Practice Q5, Q7(a))
- I can check an antiderivative by differentiating it, and explain why two correct answers can differ by a constant. (Guide: “Always check by differentiating”; Practice Q5(d))
- BC only: I can decide between substitution, integration by parts and linear partial fractions. (Guide: Worked example 3; Practice Q7(d))
All ticked? Move on to Topic 7.1, Modeling Situations with Differential Equations, or return to the Calculus AB roadmap or the Calculus BC roadmap.