MA 123, Chapter 10:
Formulas for integrals: integrals, antiderivatives, and
theFundamental Theorem of Calculus (pp. 207-233,
Gootman)
Chapter’s Goal:
• Understand the statement of the Fundamental
Theorem of Calculus.
• Learn how to compute the antiderivative of
some basic functions.
• Learn how to use the substitution method to
compute the antiderivative of morecomplex
functions.
• Learn how to solve area and distance traveled
problems by means of antiderivatives.
– p. 263/293
Example 1:
Evaluate the indefinite integral
∫
(t3 + 3t2 + 4t+ 9) dt.
– p. 264/293
Example 2:
Evaluate the indefinite integral
∫
6√tdt.
– p. 265/293
Example 3:
Evaluate the indefinite integral
∫
t3(t+ 2) dt.
– p. 266/293
Example 4:
Evaluate the indefinite integral
∫
x2 + 9
x2dx.
– p. 267/293
Example 5:
Find a formula for A(x) =
∫ x
1
(4t+ 2) dt, that is,
evaluate the definite integral of the function
f(t) = 4t+ 2 over the interval [1, x] inside [1, 10]. Find
the values A(5), A(10), A(1). What is the derivative of
A(x) with respect to x?
– p. 268/293
Example 6:
Compute the derivative of F (x) if
F (x) =
∫ x
2
(t4 + t3 + t+ 9) dt.
– p. 269/293
Example 7:
Compute the derivative of g(s) if
g(s) =
∫ s
5
8√u2 + u+ 2
du.
– p. 270/293
Example 8:
Suppose f(x) =
∫ x
1
√t2 − 7t+ 12.25 dt. For which
positive value of x does f ′(x) equal 0?
– p. 271/293
Example 9:
Find the value of x at which
F (x) =
∫ x
3
(t8 + t6 + t4 + t2 + 1) dt takes its minimum on
the interval [3, 100]. The value of x that gives a
minimum of F (x) is .
– p. 272/293
Example 10:
Find the value of x at which G(x) =
∫ x
−5
(|t|+ 2) dt
takes its maximum on the interval [−5, 100]. The value
of x that gives a maximum of G(x) is .
– p. 273/293
Example 11:
Evaluate the integral
∫ 5
0
(t2 + 1)dt.
– p. 274/293
Example 12:
Evaluate the integral
∫ −5
−7
(
1
t
)2
dt.
– p. 275/293
Example 13:
Evaluate the integral
∫ 2
0
ex dx.
– p. 276/293
Example 14:
Evaluate the integral
∫ 2
−2
(t5 + t4 + t3 + t2 + t+ 1) dt.
– p. 277/293
Example 15:
Evaluate the integral
∫ 12
−6
|t| dt.
– p. 278/293
Example 16:
Evaluate the integral
∫ 5
2
(
3u5 +7
u
)
du.
– p. 279/293
Example 17:
Suppose
f(x) =
2x, x ≤ 2;
8
x, x > 2.
Evaluate the integral
∫ 5
0
f(x) dx.
– p. 280/293
Example 18:
Evaluate the integral
∫ x
0
(t+ 9)2 dt.
– p. 281/293
Example 19:
Evaluate the integral
∫ x
0
√3t+ 7 dt.
– p. 282/293
Example 20:
Evaluate the integral
∫ x
0
1
(5t+ 4)2dt.
– p. 283/293
Example 21:
Evaluate the integral
∫ 5
0
√2t+ 1 dt.
– p. 284/293
Example 22:
Evaluate the integral
∫ 1
0
5 e5x+1 dx
– p. 285/293
Example 23:
Evaluate the integral
∫ 3
0
2x
x2 + 1dx.
– p. 286/293
Example 24:
Compute the derivative of F (x) if F (x) =
∫ x2
0
2t dt.
– p. 287/293
Example 25:
A train travels along a track and its velocity (in miles
per hour) is given by v(t) = 76t for the first half hour of
travel. Its velocity is constant and equal to v(t) = 76/2
after the first half hour. Here time t is measured in
hours. How far (in miles) does the train travel in the
second hour of travel?
– p. 288/293
Example 26:
A train travels along a track and its velocity (in miles
per hour) is given by v(t) = 76t for the first half hour of
travel. Its velocity is constant and equal to v(t) = 76/2
after the first half hour. Here time t is measured in
hours. How far (in miles) does the train travel in the first
hour of travel?
– p. 289/293
Example 27:
A rock is dropped from a height of 21 feet. Its velocity
in feet per second at time t after it is dropped is given
by v(t) = −32t where time t is measured in seconds.
How far is the rock from the ground one second after
it is dropped?
– p. 290/293
Example 28:
Suppose an object is thrown down from a cliff with an
initial speed of 5 feet per sec, and its speed in ft/sec
after t seconds is given by s(t) = 10t+ 5. If the object
lands after 7 seconds, how high (in ft) is the cliff?
– p. 291/293
Example 29:
A car is traveling due east. Its velocity (in miles per
hour) at time t hours is given by
v(t) = −2.5t2 + 10t+ 50.
How far did the car travel during the first five hours of
the trip?
– p. 292/293
Example 30:
What is the average of f(x) = x2 over the interval
[0, 6]?
– p. 293/293
Recommended
