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Unit 5 – Integration – Test 1 – Study Guide
Concepts to know:
1. Finding anti-derivatives using the reverse-chain rule (with or without u-substitution) – must know trigonometric derivatives/integrals, as well as ‘e’ and ln.
2. Calculating definite integrals (including integrals calculated using geometric formulas)
3. Identifying an integral as a limit of a Riemann sum.
4. Motion problems with integration
5. Estimating Integrals using Riemann sums (LRAM, RRAM, MRAM, and Trapezoidal approximations.)
Basic integrals that should be memorized:
Reverse Power Rule:
Critical Integrals to Know:
න𝑒𝑢𝑑𝑢 =
න1
𝑢𝑑𝑢 =
න𝑥𝑛𝑑𝑥 =
Evaluating a Definite Integral:
න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎 න
𝑎
𝑏
𝑓′ 𝑥 𝑑𝑥 = 𝑓 𝑏 − 𝑓 𝑎or
Where F is the anti-derivative of f
Definite and Indefinite Integral PracticeIf U-sub doesn’t work, try algebraic manipulation and/or simplification
න𝑒2𝑥 sin 𝑒2𝑥 − 𝑒 𝑑𝑥
Multiple Choice Released AP Questions – Definite and Indefinite Integrals
The Definite Integral as Area, and Integrals Using Geometric Formulas
න
−4
0
16 − 𝑥2𝑑𝑥 න
2 2
−2 2
8 − 𝑥2 + 3 𝑑𝑥
The Definite Integral a Limit of a Riemann Sum
Limit Statement Definite Integral
lim𝑛→∞
𝑘=1
𝑛
(2 +3
𝑛𝑘)2+2 ∙
3
𝑛
lim𝑛→∞
𝑘=1
𝑛
3(2
𝑛(𝑘 − 1) + 6 ∙
2
𝑛
lim𝑛→∞
𝑘=1
𝑛
2(4 +5
𝑛𝑘)3 + 6 ∙
5
𝑛
lim𝑛→∞
𝑘=1
𝑛
4 − (1
𝑛𝑘)2 ∙
1
𝑛
lim𝑛→∞
𝑘=1
𝑛
sin(3 +7
𝑛(𝑘 − 1) ∙
7
𝑛
Definite Integral Limit Statement
න−1
2
(𝑥2 − 6)𝑑𝑥
න1
9
(5𝑥 + 4)𝑑𝑥
Motion Problems Utilizing Integration
Rectangular and Trapezoidal Approximations