Calculus 4.1, 4.3, 4.4 Indefinite and Definite Integration
Review Worksheet
Evaluate the integral.
1.
∫ 6 dx
2.
∫ (3x
3.
∫x
4.
∫
5.
6.
1
3
4
− 2 x 2 + 5) dx
3
dx
t dt
∫
x3 + x
dx
x
∫
3 + 4x
3
2
dx
x
7.
∫ 5 sec x tan x
8.
sin 3 θ
∫ 1 − cos 2 θ dθ
9.
∫ 3 csc
10.
sec 3 θ tan θ
∫ 1 + tan 2 θ dθ
11.
∫ 3 csc x cot x
12.
cos 3 θ
∫ 2 − 2 sin 2 θ dθ
13.
Find the particular solution of the equation f ' ( x) = 4 x −1 2 that satisfies the condition
f(1) = 6.
Scherer
2
dx
x dx
dx
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14.
Find y = f(x) if f''(x) = x2 , f'(0) = 7, and f(0) = 2.
15.
Use a (t ) = −32 ft / sec 2 as the acceleration due to gravity. A ball is thrown vertically
upward from the ground with an initial velocity of 96 feet per second. How high will it
go?
16.
Use a (t ) = −32 ft / sec 2 as the acceleration due to gravity. A ball is thrown vertically
upward from the ground with an initial velocity of 56 feet per second. For how many
seconds will the ball be going upward?
17.
An object has a constant acceleration of 72 feet per second squared, an initial velocity of
17 feet per second, and an initial position of 10 feet. Find the position function
describing the motion of this object.
18.
Scherer
3
dP
= 50t 2 − 100t 2 where P is
dt
the population size and t is the time in years. The initial population is 25,000. Find the
population function. Then use a graphing utility to graph the function, and then use the
graph to estimate how many years it will take for the population to reach 50,000.
The rate of growth of a particular population is given by
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19.
Write the definite integral that represents the area of the region enclosed by
y = 4 x − x 2 and the x-axis.
20.
Write the definite integral for the area of the region lying in the upper half of the ellipse
given by 4 x 2 + y 2 = 4 .
Evaluate the following definite integrals.
4
21.
∫
x dx
1
2
22.
∫ (2 x − 1)dx
−1
2
23.
1
∫x
2
dx
1
1
24.
∫(
3
t − 2)dt
−1
1
25.
∫x
dx
−1
5π
26.
∫ sin xdx
π
Scherer
4
2
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3π
27.
4
∫ (− csc
π
2
t )dt
4
5 + 6x + x 2
∫3 5 + x dx
5
28.
π
29.
2
∫ cos xdx
0
30.
Find the average value of f ( x) = 2 x 2 + 3 on the interval [0,2].
31.
⎡π π ⎤
Find the average value of f ( x) = sin x on the interval ⎢ , ⎥ .
⎣4 2⎦
x
1
dt find F ' ( x) and F ' (2) .
1+ t4
2
32.
Given F ( x) = ∫
33.
Evaluate
34.
d
Evaluate
dx
x
35.
d
(2t 2 + 5) 2 dt
∫
dx 5
2 x2
∫ (4t + 1)
3
dt
3
Find the value of c guaranteed by the Mean Value Theorem for Integrals for f ( x) =
on the interval [1,4].
Scherer
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4
x2