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 what ∫ f(x) dx = F(x) + C means and why the + C is needed. (Guide: “From one antiderivative to a family”; Practice Q6(b))
- I can describe the graphs of a family of antiderivatives as vertical shifts with equal slopes at each x. (Guide: Figure 1; Practice Q6(c))
- I can explain the difference between an indefinite integral and a definite integral. (Guide: comparison table; Practice Q6(d))
- I can explain why some functions, such as e^(x²), have no formula antiderivative, yet still have antiderivatives. (Guide: Worked example 3(c); Practice Q7(d))
Skills
- I can integrate powers, including negative and fractional powers, and use ln|x| for 1/x. (Guide: basic rules table; Practice Q1, Q2)
- I can integrate eˣ, aˣ, sin x, cos x, sec²x, csc²x, sec x tan x and csc x cot x with the correct signs. (Guide: Worked example 2; Practice Q3, Q7(c))
- I can recognise 1/√(1 − x²) and 1/(1 + x²) as derivatives of arcsin x and arctan x. (Guide: Worked example 2(b); Practice Q3)
- I can expand, split or use an identity to rewrite an integrand before integrating. (Guide: Worked example 1; Practice Q5, Q7(a), Q7(b))
Reasoning
- I can check any claimed antiderivative by differentiating it, and explain what is wrong when the check fails. (Guide: Worked example 3; Practice Q4, Q7(d))
- I can write integral notation correctly, with dx and + C, and without false equal signs. (Guide: “A note on notation” and “Common misconceptions”)
All ticked? Move on to Topic 6.9, Integrating Using Substitution, or return to the Calculus AB roadmap or the Calculus BC roadmap.