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 6.7 to 6.10) are on the Calculus BC roadmap.
Understanding
- I can explain partial fractions as adding fractions backwards. (Guide: “The idea: adding fractions backwards”)
- I can state when the method applies: a proper fraction whose bottom factors into different linear factors. (Guide: “When the method applies”; Practice Q6(a))
- I can recognise when substitution is quicker than partial fractions. (Guide: Common misconceptions)
Skills
- I can decompose a fraction with two linear factors, finding A and B by substituting the roots. (Guide: “Finding the constants”; Practice Q1)
- I can find the same constants by matching coefficients, and check by recombining. (Guide: “Finding the constants”; Practice Q5)
- I can decompose a fraction with three different linear factors. (Practice Q5)
- I can integrate each piece as a logarithm, keeping absolute values. (Guide: Worked example 1; Practice Q2)
- I can handle a factor such as 2x + 1, including the chain-rule factor 1/2. (Guide: “A linear factor that is not monic”; Practice Q4)
- I can divide first when the top’s degree is too high, then decompose the remainder. (Guide: “A top that is too big”; Practice Q6)
- I can evaluate a definite integral exactly using partial fractions and log laws. (Guide: Worked example 2; Practice Q3, Q6(d))
Reasoning
- I can check that no root of the denominator lies between the limits before evaluating a definite integral. (Guide: Common misconceptions; Practice Q6(d))
- I can set up and evaluate an accumulation problem with a rational rate, and state the answer with units. (Guide: Worked example 2; Practice Q7)
- I can use a definite integral of a rate, plus a starting amount, to justify a claim about a total. (Practice Q7(c))
All ticked? Move on to Topic 6.13, Evaluating Improper Integrals, or return to the Calculus BC roadmap.