AP Calculus BC - Live Review Session 4 - Integration Techniques in AP Calculus BC PDF

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Cram Sheet by fiveable; things to know for the Calculus AP test...

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Calc Calculu ulu uluss BC - 20 202 21 AP L Liv iv ive eR Rev ev evie ie iew w SSes es essio sio sion n4 Int Integr egr egrat at ation ion Te Tech ch chn niqu ique es in AP Ca Calc lc lcul ul ulus us B BC C 𝐅𝐑𝐐 𝟏 Let 𝑓(𝑥) =

𝑥2

1 where 𝑘 is a constant. − 𝑘𝑥 + 16

𝐚) Let 𝑘 = 10. Find ∫ 𝑓(𝑥)𝑑𝑥.

∞

𝐛) Let 𝑘 = 8. Find ∫ 𝑓(𝑥)𝑑𝑥 or show that it diverges. 6

𝐜) There is a value of 𝑘 such that ∫

0 𝑓 ′ (𝑥)

−3

𝐝) Let 𝑘 = 0. Find ∫

4𝑥𝑒 𝑥

2

𝑓(𝑥)

𝑓(𝑥)

1 𝑑𝑥 = ln ( ) . Find 𝑘. 16

𝑑𝑥

AP Calculus BC: Integration Techniques in AP Calculus BC

2021 AP Live

Bryan Passwater & Tony Record

𝐅𝐑𝐐 𝟐 The function 𝑓 is twice differentiable for all values of 𝑥. A portion of the graph of 𝑓 is shown above along with the line tangent to the graph of 𝑓 at 𝑥 = 12. The graph of 𝑓 has horizontal tangent lines at 𝑥 = 3, 𝑥 = 6, and 𝑥 = 10. The region R is bounded by the graph of 𝑓 and the 𝑥-axis and has area 22. 12

𝐚) Find ∫ 3𝑥𝑓 ′ (𝑥)𝑑𝑥 6

12

𝐛) Find ∫ 𝑥𝑓 ′′(𝑥)𝑑𝑥. 6

𝐜) Let 𝑘 be a constant such that ∫

1 2

0

𝑘 1 12 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥. Find 𝑘. Show the work that leads to your answer. −1 2 6

𝑥2

𝑥

3 lim 𝑓(𝑥) or show that the limit does not exist. 2𝑡 𝑑𝑡 . Find 𝑥→∞ −2 𝑒

𝐝) For 𝑥 > 16, the function 𝑓(𝑥) = ∫

AP Calculus BC: Integration Techniques in AP Calculus BC

2021 AP Live

Bryan Passwater & Tony Record

𝟓 𝐟𝐨𝐫 𝟓: 𝐌𝐂 𝐏𝐫𝐚𝐜𝐭𝐢𝐜𝐞 𝐟𝐨𝐫 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐓𝐞𝐜𝐡𝐧𝐢𝐪𝐮𝐞𝐬 𝟏. Using the substitution 𝑢 = 2𝑥 − 1, which of the following is equivalent to ∫ (A) 2 ∫

13

5

𝟐. ∫

3

0

1

2𝑥 − 1

1 𝑢−2

25

𝑑𝑢

5

(B) ∫ 𝑢−2 𝑑𝑢

1 1 − (C) ∫ 𝑢 2 𝑑𝑢 2 9

1 (B) ln(5) 2

26 (C) − 25

1

9

𝑑𝑥 ? √2𝑥 − 1

1 13 −1 (D) ∫ 𝑢 2 𝑑𝑢 2 5

𝑑𝑥 is

(A) ln(5)

𝟑. ∫

25

13

8𝑥

(𝑥 + 3)(𝑥 − 1)

(D) nonexistent

𝑑𝑥 =

(A) 4 ln|(𝑥 + 3)(𝑥 − 1)| + C (B) 12 ln|(𝑥 + 3)(𝑥 − 1)| + C (C) 2 ln|𝑥 + 3| + 6 ln|𝑥 − 1| + C (D) 6 ln|𝑥 + 3| + 2 ln|𝑥 − 1| + C

AP Calculus BC: Integration Techniques in AP Calculus BC

2021 AP Live

Bryan Passwater & Tony Record

𝟒. If ∫ 2𝑥𝑓 ′′ (𝑥)𝑑𝑥 = −4𝑥 cos ( 𝑥 (A) −2cos ( ) 2

𝟓. ∫

∞

2

(A)

𝑥

2

𝑥 ) + C, then which of the following could be 𝑓(𝑥)? ) + 8 sin ( 2

𝑑𝑥 is 𝑥2 + 4 𝜋 8

(B)

𝑥 (B) − 4 sin ( ) 2

𝜋 4

𝑥 (C) 4 cos ( ) 2

(C)

AP Calculus BC: Integration Techniques in AP Calculus BC

𝜋 2

2021 AP Live

𝑥) (D) 8 sin (2

(D) nonexistent

Bryan Passwater & Tony Record...