264 Chapter 4: Integration
4.5 Integration by Substitution
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the indefinite integral .
a.
b.
c.
d.
e.
____ 2. Find the indefinite integral .
a.
b.
c.
d.
e.
4.5 Integration by Substitution 265
____ 3. Find the indefinite integral .
a.
b.
c.
d.
e.
____ 4. Find the indefinite integral of the following function and check the result by
differentiation.
a.
b.
c.
d.
e.
266 Chapter 4: Integration
____ 5. Find the indefinite integral of the following function and check the result by
differentiation.
a.
b.
c.
d.
e.
____ 6. Find the indefinite integral of the following function and check the result by
differentiation.
a.
b.
c.
d.
e.
____ 7. Solve the differential equation.
a.
b.
c.
4.5 Integration by Substitution 267
d.
e.
____ 8. Solve the differential equation.
a.
b.
c.
d.
e.
____ 9. Find the indefinite integral of the following function.
a.
b.
c.
d. + C
e.
268 Chapter 4: Integration
____ 10. Find the indefinite integral of the following function.
a.
b.
c.
d.
e.
____ 11. Find the indefinite integral .
a.
b.
c.
d.
e.
____ 12. Find the indefinite integral of the following function.
a.
b.
c.
d.
e.
4.5 Integration by Substitution 269
____ 13. Evaluate the following definite integral.
Use a graphing utility to check your answer.
a.
b.
c.
d.
e.
____ 14. The rate of depreciation of a machine is inversely proportional to the square
of where V is the value of the machine t years after it was purchased. The initial value of the
machine was and its value decreased in the first year. Estimate its value after 3
years. Round your answer to the nearest integer.
a. $500,000
b. $175,000
c. $275,000
d. $650,000
e. $550,000
____ 15. The sales S (in thousands of units) of a seasonal product are given by the
model where t is the time in months, with corresponding to January.
Find the average sales for the first quarter . Round your answer to three decimal places.
a. 131.598 thousand units
b. 92.442 thousand units
c. 74.859 thousand units
d. 102.479 thousand units
e. 92.924 thousand units
270 Chapter 4: Integration
____ 16. The oscillating current in an electrical circuit is where I
is measured in amperes and t is measured in seconds. Find the average current for the time interval
. Round your answer to three decimal places.
a. 2.273 amps
b. 1.273 amps
c. 0.637 amps
d. 2.546 amps
e. 1.637 amps
____ 17. Suppose that the probability that a person will remember between 100a% and 100b%
of material learned in an experiment is where x represents the proportion
remembered. Determine from the figure below, the probability that a randomly chosen individual will
recall between 50% and 55% of the material? Express your answer as a percent rounded to three
decimal places.
a. 39.000%
b. 6.800%
c. 1.600%
d. 47.800%
e. 85.300%
____ 18. A manufacturer of fertilizer finds that national sales of fertilizer follow the seasonal
pattern where F is measured in pounds and t represents the time in
days, with corresponding to January 1. The manufacturer wants to set up a schedule to produce a
uniform amount of fertilizer each day. What should this amount be?
a. 600,000 lbs
b. 300,000 lbs
c. 2,400,000 lbs
d. 150,000 lbs
e. 1,200,000 lbs
4.5 Integration by Substitution 271
4.5 Integration by Substitution
Answer Section