Integration - Lamar University calculus/hoffman/ch05sec01.pdfx, and C is called the constant of...

Chapter� 1. Antidifferentiation:

The Indefinite Integral

� 2. Integration bySubstitution

� 3. Introduction toDifferential Equations

� 4. Integration by Parts

� Chapter Summary andReview Problems

Integration

372 Chapter 5 Integration

In many problems, the derivative of a function is known, and the goal is to find thefunction itself. For example, a sociologist who knows the rate at which the popula-tion is growing may wish to use this information to predict future population levels;a physicist who knows the speed of a moving body may wish to calculate the futureposition of the body; an economist who knows the rate of inflation may wish to esti-mate future prices.

The process of obtaining a function from its derivative is called antidifferentia-tion or indefinite integration.

Sometimes we will write the equation F�(x) � f(x) as � f(x).

Later in this section, you will learn techniques you can use to find antideriva-tives. Once you have found what you believe to be an antiderivative of a function,you can always check your answer by differentiating. You should get the original func-tion back. Here is an example.

Verify that F(x) � x3 � 5x � 2 is an antiderivative of f(x) � x2 � 5.

SolutionF(x) is an antiderivative of f(x) if and only if F�(x) � f(x). Differentiate F and youwill find that

F�(x) �

� x2 � 5 � f(x)

as required.

1

3�3x2� � 5

1

3

dF

dx

Antiderivative � A function F(x) for which

F�(x) � f(x)

for every x in the domain of f is said to be an antiderivative of f(x).

Antidiffer-entiation:

The IndefiniteIntegral

1

Note

EXAMPLE 1 .1EXAMPLE 1 .1

A function has more than one antiderivative. For example, one antiderivative of thefunction f(x) � 3x2 is F(x) � x3, since

F�(x) � 3x2 � f(x)

but so are x3 � 12 and x3 � 5 and x3 � �, since

(x3 � 12) � 3x2, (x3 � 5) � 3x2, (x3 � �) � 3x2

In general, if F is one antiderivative of f, then any function G of the general formG(x) � F(x) � C for constant C is also an antiderivative of f. Conversely, it can beshown that if F and G are both antiderivatives of f, then G(x) � F(x) � C, for someconstant C (see Problem 45). To summarize:

There is a simple geometric interpretation for the fact that any two antiderivativesof the same continuous function f differ by at most a constant. When we say that Fand G are both antiderivatives of f, we mean that F�(x) � G�(x) � f(x), so the slope(that is, the derivative) of the curve y � F(x) for each value of x is the same as theslope of y � G(x) at x. In other words, the graph of G(x) is a vertical translation ofthe graph of F(x), as indicated in Figure 5.1 for the antiderivative of f(x) � 3x2.

FIGURE 5.1 Some antiderivatives of f(x) � 3x2.

x

yy = x3 + π

y = x3 – 5

y = x3

Fundamental Property of Antiderivatives � If F(x) is anantiderivative of the continuous function f(x), then any other antiderivative off(x) has the form G(x) � F(x) � C for some constant C.

d

dx

d

dx

d

dx

THE GENERAL ANTIDERIVATIVEOF A FUNCTION

Chapter 5 � Section 1 Antidifferentiation: The Indefinite Integral 373

You have just seen that if F(x) is one antiderivative of the continuous function f(x),then all such antiderivatives are characterized by F(x) � C for constant C. We willrepresent the family of all antiderivatives of f(x) by using the symbolism

f(x) dx � F(x) � C

which is called the indefinite integral of f.

In this context, the symbol is called the integral sign, f(x) is called the inte-

grand, dx indicates that the antidifferentiation process is with respect to the variablex, and C is called the constant of integration. For instance, the indefinite integral off(x) � 3x2 is

3x2 dx � x3 � C

The integral symbol resembles an elongated “s,” which stands for “sum.” In Chap-

ter 6, you will see a surprising connection between antiderivatives and sums that isso important it is known as the fundamental theorem of calculus.

To check an antidifferentiation calculation, differentiate your answer F(x) � C.If the derivative equals f(x), your calculation is correct, but if it is anything otherthan f(x), you’ve made a mistake. This connection between differentiation and anti-differentiation enables us to use familiar rules for derivatives to establish the follow-ing analogous rules for antiderivatives.

��

�

�

THE INDEFINITE INTEGRAL


Rules for Integrating Common Functions

The constant rule: k dx � kx � C for constant k

The power rule: for all n � �1

The logarithmic rule: for all x � 0

The exponential rule: for constant k � 0� ekxdx �

1

kekx � C

� 1

xdx � ln |x| � C

�

xndx �1

n � 1xn�1 � C

�

Store the function F(x) � x3

into Y1 of the equation editor in

bold graphing style. Generate a

family of vertical transforma-

tions Y1 � c, for c � �4, �2,

2, 4, placed into Y2 to Y5, with

the window [�4, 4]1 by [�5,

5]1. At x � 1, what do you ob-

serve about the slopes for all of

these curves?

To verify the power rule, it is enough to show that the derivative of xn�1

is xn:

For the logarithmic rule, if x � 0, then �x� � x and

If x 0, then �x � 0 and ln �x� � ln (�x), and it follows from the chain rule that

Thus, for all x � 0,

so

You are asked to verify the constant rule and exponential rule in Problem 46.

Notice that the logarithm rule “fills the gap” in the power rule; namely, thecase where n � �1. You may wish to blend the two rules into the follow-ing combined form:

Find the following integrals:

(a) (b) (c) (d) �e�3xdx� 1

�xdx�x17dx�3dx

�xndx � �xn�1

n � 1� C

ln |x| � C

if n � �1

if n � �1

�1

xdx � ln |x| � C

d

dx(ln |x|) �

1

x

d

dx(ln |x|) �

d

dx[ln (�x)] �

1

(�x)(�1) �

1

x

d

dx(ln |x|) �

d

dx(ln x) �

1

x

d

dx�1

n � 1xn�1 �

1

n � 1(n � 1) xn] � xn

1

n � 1


Note


Solution

(a) Use the constant rule with k � 3: � 3x � C

(b) Use the power rule with n � 17:

(c) Use the power rule with n � � : Since n � 1 � ,

(d) Use the exponential rule with k � �3:

Example 1.2 illustrates how certain basic functions can be integrated, but whatabout combinations of functions, such as the polynomial x5 � 2x3 � 7 or an expres-sion like 5e�x � ? The following algebraic rules enable you to handle suchexpressions in a natural fashion.

To prove the constant multiple rule, note that if � f(x), then

[kF(x)] � k � kf(x)

which means that

dF

dx

d

dx

dF

dx

�x

�e�3xdx �1

(�3)e�3x � C

� dx

�x� �x�1/2dx �

1

(1/2)x1/2 � C � 2�x � C

1

2

1

2

�x17dx �1

18x18 � C

�3dx


Algebraic Rules for Indefinite Integration

The constant multiple rule: kf(x) dx � k f(x) dx for constant k

The sum rule: [ f(x) � g(x)] dx � f(x) dx � g(x) dx

The difference rule: [ f(x) � g(x)] dx � f(x) dx � g(x) dx��

��

Graph the function F(x) � ln �x�in bold and f(x) � 1/x in regular

graphing style, using the deci-

mal window [�4.7, 4.7]1 by

[�3.1, 3.1]1. Confirm that at

any point x � 0, the derivative

of F(x) is identical in value to

f(x) at that particular point.

kf(x) dx � k f(x) dx

The sum and difference rules can be established in a similar fashion.

Find the following integrals:

(a) (2x5 � 8x3 � 3x2 � 5) dx

(b)

(c)

Solution

(a) By using the power rule in conjunction with the sum and difference rules and themultiple rule, you get

(b) There is no “quotient rule” for integration, but you can still divide the denomi-nator into the numerator and then integrate using the method in part (a):

(c)

� 3� 1

�5e�5t� �

1

(3/2)t3/2 � C �

�3

5e�5t �

2

3t3/2 � C

�(3e�5t � �t ) dt � �(3e�5t � t1/2) dt

�1

3x3 � 2x � 7 ln |x| � C

��x3 � 2x � 7

x � dx � ��x2 � 2 �7

x� dx

�1

3x6 � 2x4 � x3 � 5x � C

� 2�x6

6 � � 8�x4

4 � � 3�x3

3 � � 5x � C

�(2x5 � 8x3 � 3x2 � 5) dx � 2�x5dx � 8�x3dx � 3�x2dx � �5 dx

�(3e�5t � �t) dt

��x3 � 2x � 7

x �dx

�

��



Find the function f(x) whose tangent has slope 3x2 � 1 for each value of x and whosegraph passes through the point (2, 6).

SolutionThe slope of the tangent at each point (x, f(x)) is the derivative f �(x). Thus,

f �(x) � 3x2 � 1

and so f(x) is the antiderivative

To find C, use the fact that the graph of f passes through (2, 6). That is, substi-tute x � 2 and f(2) � 6 into the equation for f(x) and solve for C to get

6 � (2)3 � 2 � C or C � �4

That is, the desired function is f(x) � x3 � x � 4.

Here are three problems in which the rate of change of a quantity is known and the goalis to find an expression for the quantity itself. Since the rate of change is the derivativeof the quantity, you find the expression for the quantity itself by antidifferentiation.

A manufacturer has found that marginal cost is 3q2 � 60q � 400 dollars per unitwhen q units have been produced. The total cost of producing the first 2 units is $900.What is the total cost of producing the first 5 units?

SolutionRecall that the marginal cost is the derivative of the total cost function C(q). Thus,

C�(q) � 3q2 � 60q � 400

and so C(q) must be the antiderivative

for some constant K. (The letter K was used for the constant to avoid confusion withthe cost function C.)

C(q) � �C�(q) dq � �(3q2 � 60q � 400) dq � q3 � 30q2 � 400q � K

PRACTICAL APPLICATIONS

f(x) � � f�(x) dx � �(3x2 � 1) dx � x3 � x � C



EXAMPLE 1 .4EXAMPLE 1 .4Graph in bold the function

f(x) � 3x2 � 1 from Example

1.4, along with the family of an-

tiderivatives F(x) � x3 � x �

L1, where L1 is a list of integer

values from �5 to 5. Use the

window [0, 2.35].5 by [�2,

12]1. Which of these antideriv-

atives passes through (2, 6)?

Repeat this exercise for f(x) �

3x2 � 2.

The value of K is determined by the fact that C(2) � 900. In particular,

900 � (2)3 � 30(2)2 � 400(2) � K or K � 212

Hence, C(q) � q3 � 30q2 � 400q � 212

and the cost of producing the first 5 units is

C(5) � (5)3 � 30(5)2 � 400(5) � 212 � $1,587

It is estimated that x months from now the population of a certain town will be chang-ing at the rate of 2 � people per month. The current population is 5,000. Whatwill be the population 9 months from now?

SolutionLet P(x) denote the population of the town x months from now. Then the rate ofchange of the population with respect to time is the derivative

It follows that the population function P(x) is an antiderivative of 2 � . That is,

for some constant C. To determine C, use the fact that at present (when x � 0) thepopulation is 5,000. That is,

5,000 � 2(0) � 4(0)3/2 � C or C � 5,000

Hence, P(x) � 2x � 4x3/2 � 5,000

and the population 9 months from now will be

P(9) � 2(9) � 4(27) � 5,000 � 5,126

A retailer receives a shipment of 10,000 kilograms of rice that will be used up overa 5-month period at the constant rate of 2,000 kilograms per month. If storage costsare 1 cent per kilogram per month, how much will the retailer pay in storage costsover the next 5 months?

P(x) � �dP

dx dx � �(2 � 6�x) dx � 2x � 4x3/2 � C

6�x

dP

dx� 2 � 6�x

6�x




Graph the function P(x) from

Example 1.6, using the window

[0, 47]5 by [4,000, 7,000]500.

Display the population 9 months

from now. When will the popu-

lation hit 6,000 people?

SolutionLet S(t) denote the total storage cost (in dollars) over t months. Since the rice is usedup at a constant rate of 2,000 kilograms per month, the number of kilograms of ricein storage after t months is 10,000 � 2,000t. Therefore, since storage costs are 1 centper kilogram per month, the rate of change of the storage cost with respect to timeis

It follows that S(t) is an antiderivative of

0.01(10,000 � 2,000t) � 100 � 20t

That is,

for some constant C. To determine C, use the fact that at the time the shipment arrives(when t � 0) there is no cost, so that

0 � 100(0) � 10(0)2 � C or C � 0

Hence, S(t) � 100t � 10t2

and the total storage cost over the next 5 months will be

S(5) � 100(5) � 10(5)2 � $250

Recall from Section 2 of Chapter 2 that if an object moves along a straight line with

displacement s(t), then its velocity is given by v � and its acceleration by a � .

If the acceleration of the object is given, then its velocity and displacement can be found by integration. Here is an example.

After its brakes are applied, a certain car decelerates at the constant rate of 22 feetper second per second. If the car is traveling at 45 miles per hour (66 feet per sec-ond) when the brakes are applied, how far does it travel before coming to a completestop?

SolutionLet s(t) denote the displacement (position) of the car t seconds after the brakes are

dv

dt

ds

dt

MOTION ALONG A LINE

� 100t � 10t2 � C

S(t) � �dS

dt dt � �(100 � 20t) dt

dS

dt� �monthly cost

per kilogram��number ofkilograms� � 0.01(10,000 � 2,000t)



applied. Inasmuch as the car decelerates at 22 feet per second, it follows that a(t) ��22 (the negative sign indicates that the car is slowing down), and

Integrating, you find that the velocity at time t is given by

To evaluate C1, note that v � 66 when t � 0 so that

66 � v(0) � �22(0) � C1

and C1 � 66. Thus, the velocity at time t is v(t) � �22t � 66.

Next, to find the displacement s(t), begin with the fact that

and use integration to show that

Since s(0) � 0 (do you see why?), it follows that C2 � 0 and

s(t) � �11t2 � 66t

Finally, to find the stopping distance, note that the car stops when v(t) � 0, and thisoccurs when

v(t) � �22t � 66 � 0

Solving this equation, you find that the car stops after 3 seconds of deceleration, andin that time it has traveled

s(3) � �11(3)2 � 66(3) � 99 feet

In Problems 1 through 20, find the indicated integral. Check your answers bydifferentiation.

1. 2. �x3/4 dx�x5 dx

s(t) � �(�22t � 66) dt � � 11t2 � 66t � C2

ds

dt� v(t) � �22t � 66

v(t) � ��22 dt � �22t � C1

dv

dt� a(t) � �22


P . R . O . B . L . E . M . S 5.1P . R . O . B . L . E . M . S 5.1

Refer to Example 1.8. Change

the variable t to x and graph the

position function s(x) using the

window [0, 9.4]1 by [0, 200]10.

Locate the stopping time and

the corresponding position on

the graph. Work the problem

again for v(0) � 60 mph (88

feet per second). What is hap-

pening at the 3-second mark?

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. Find the function whose tangent has slope 4x � 1 for each value of x and whosegraph passes through the point (1, 2).

22. Find the function whose tangent has slope 3x2 � 6x � 2 for each value of x andwhose graph passes through the point (0, 6).

23. Find the function whose tangent has slope x3 � � 2 for each value of x and

whose graph passes through the point (1, 3).

24. Find a function whose graph has a relative minimum when x � 1 and a relativemaximum when x � 4.

POPULATION GROWTH 25. It is estimated that t months from now the population of a certain town will bechanging at the rate of 4 � 5t2/3 people per month. If the current population is10,000, what will be the population 8 months from now?

DISTANCE AND VELOCITY 26. An object is moving so that its velocity after t minutes is v(t) � 1 � 4t � 3t2 metersper minute. How far does the object travel during the third minute?

DISTANCE AND VELOCITY 27. An object is moving so that its velocity after t minutes is v(t) � 3 � 2t � 6t2 metersper minute. How far does the object travel during the second minute?

2

x2

�x(2x � 1)2 dx��t(t2 � 1) dt

�y3�2y �1

y� dy�(x3 � 2x2)�1

x� 5� dx

�x2 � 3x � 2

�x dx��x2 � 2x � 1

x2 � dx

��2eu �6

u� ln 2� du�� 1

3u�

3

2u2 � e2 ��u

2 � du

��x3 �1

2�x� �2� dx��ex

2� x�x� dx

�� 1

2y�

2

y2 �3

�y� dy��3�y �2

y3 �1

y� dy

�(x1/2 � 3x2/3 � 6) dx�(3t2 � �5t � 2) dt

�3ex dx�5 dx

��t dt� 1

x2 dx


LAND VALUES 28. It is estimated that t years from now the value of a certain parcel of land will beincreasing at the rate of V�(t) dollars per year. Find an expression for the amount bywhich the value of the land will increase during the next 5 years.

ADMISSION TO EVENTS 29. The promoters of a county fair estimate that t hours after the gates open at 9:00 A.M.,visitors will be entering the fair at the rate of N�(t) people per hour. Find an expressionfor the number of people who will enter the fair between 11:00 A.M. and 1:00 P.M.

STORAGE COST 30. A retailer receives a shipment of 12,000 pounds of soybeans that will be used at aconstant rate of 300 pounds per week. If the cost of storing the soybeans is 0.2 centper pound per week, how much will the retailer have to pay in storage costs over thenext 40 weeks?

WATER POLLUTION 31. It is estimated that t years from now the population of a certain lakeside communitywill be changing at the rate of 0.6t2 � 0.2t � 0.5 thousand people per year.Environmentalists have found that the level of pollution in the lake increases at therate of approximately 5 units per 1,000 people. By how much will the pollution inthe lake increase during the next 2 years?

AIR POLLUTION 32. An environmental study of a certain community suggests that t years from now thelevel of carbon monoxide in the air will be changing at the rate of 0.1t � 0.1 partsper million per year. If the current level of carbon monoxide in the air is 3.4 parts permillion, what will be the level 3 years from now?

MARGINAL COST 33. A manufacturer has found that marginal cost is 6q � 1 dollars per unit when q unitshave been produced. The total cost (including overhead) of producing the first unitis $130. What is the total cost of producing the first 10 units?

MARGINAL PROFIT 34. A manufacturer estimates marginal revenue to be 100q�1/2 dollars per unit when thelevel of production is q units. The corresponding marginal cost has been found to be0.4q dollars per unit. Suppose the manufacturer’s profit is $520 when the level ofproduction is 16 units. What is the manufacturer’s profit when the level ofproduction is 25 units?

MARGINAL PROFIT 35. The marginal profit (the derivative of profit) of a certain company is 100 � 2qdollars per unit when q units are produced. If the company’s profit is $700 when 10units are produced, what is the company’s maximum possible profit?

MARGINAL REVENUE 36. Suppose it has been determined that the marginal revenue associated with theproduction of x units of a particular commodity is R�(x) � 240 � 4x dollars per unit.What is the revenue function R(x)? You may assume R(0) � 0. What price will bepaid for each unit when the level of production is x � 5 units?

FLOW OF BLOOD 37. One of Poiseuille’s laws for the flow of blood in an artery says that if v(r) is thevelocity of flow r cm from the central axis of the artery, then the velocity decreasesat a rate proportional to r. That is,

v�(r) � �ar


where a is a positive constant.* Find an expression for v(r). Assume v(R) � 0,where R is the radius of the artery.

TREE GROWTH 38. An environmentalist finds that a certain type of tree grows in such a way that itsheight h(t) after t years is changing at the rate of

h�(t) � 0.2t2/3 � ft/yr

If the tree was 2 feet tall when it was planted, how tall will it be in 27 years?

ENDANGERED SPECIES 39. A conservationist finds that the population P(t) of a certain endangered species isgrowing at the rate of P�(t) � 0.51e�0.03t, where t is the number of years afterrecords began to be kept.(a) If the population is P0 � 500 now (at time t � 0), what will it be in 10 years?(b) Read an article on endangered species and write a paragraph on the use of

mathematical models in studying populations of such species.†

EFFECT OF A TOXIN 40. A toxin is introduced to a bacterial colony, and t hours later, the population P(t) ofthe colony is changing at the rate

If there were 1 million bacteria in the colony when the toxin was introduced, whatis P(t)? [Hint: Recall that 3x � ex ln 3.]

LEARNING CURVE 41. Let f(x) represent the total number of items a subject has memorized x minutes afterbeing presented with a long list of items to learn. Psychologists refer to the graph ofy � f(x) as a learning curve and f�(x) as the learning rate. The time of peak efficiencyis the time when the learning rate is maximized. Suppose the learning rate is

f�(x) � 0.1(10 � 12x � 0.6x2) for 0 x 25

(a) What is the learning rate at peak efficiency?(b) What is f(x)?(c) What is the largest number of items memorized by the subject?

dP

dt� �(ln 3)34�t

�t

PROBLEM 37

Rr

Artery


* E. Batschelet, Introduction to Mathematics for Life Scientists, 2nd ed., Springer-Verlag, New York,1976, pages 101–103.

† You may wish to begin your research with the journal Ecology.

ENERGY EXPENDITURE 42. In a paper written in 1971,* V. A. Tucker and K. Schmidt-Koenig investigated therelationship between the velocity v (km/hr) of a bird in flight and the energy E(v)expended by the bird. Their studies showed that for a certain kind of parakeet, therate of change of the energy with respect to velocity is given by

Suppose it is known that when the parakeet flies at the most economical velocity(the velocity that minimizes E), the energy expended is Emin � 0.6 (calorie pergram mass per kilometer). What is E(v)?

SPY STORY 43. Our spy, intent on avenging the death of Siggy Leiter (Problem 43 in Exercise Set4.2), is driving a sports car toward the lair of the fiend who killed his friend. Toremain as inconspicuous as possible, he is traveling at the legal speed of 60 mph (88feet per second) when suddenly, he sees a camel in the road, 199 feet in front of him.It takes him 00.7 seconds to react to the crisis; then he hits the brakes, and the cardecelerates at the constant rate of 28 ft/sec2 (28 feet per second, per second). Doeshe stop before hitting the camel?

MARGINAL PROPENSITY 44. Suppose the consumption function for a particular country is c(x), where x isTO CONSUME national disposable income. Then the marginal propensity to consume is c�(x).

Suppose

c�(x) � 0.9 � 0.3

and consumption is 10 billion dollars when x � 0. Find c(x).

45. If H�(x) � 0 for all real numbers x, what must be true about the graph of H(x)?Explain how your observation can be used to show that if F�(x) � G�(x) for allx, then F(x) � G(x) � C.

46. (a) Prove the constant rule:

(b) Prove the exponential rule:

47. What is bx dx for base b (b � 0, b � 1)? [Hint: Recall that bx � ex ln b.]

48. It is estimated that x months from now, the population of a certain town will bechanging at the rate of P�(x) � 2 � 1.5 people per month. The current populationis 5,000.

�x

��ekx dx �

1

kekx � C.

�k dx � kx � C.

�x

dE

dv�

0.074v2 � 112.65

v2 v � 0


* E. Batschelet, Introduction to Mathematics for Life Scientists, 2nd ed., Springer-Verlag, New York,1976, pages 101–103.

(a) Find a function P(x) that satisfies these conditions. Use the graphing utility ofyour calculator to graph this function.

(b) Use trace and zoom to determine the level of population 9 months from now.When will the population be 7,590?

(c) Suppose the current population were 2,000 (instead of 5,000). Sketch the graphof P(x) with this assumption. Then sketch the graph of P(x) assuming currentpopulations of 4,000 and 6,000. What is the difference between the graphs?

49. A car traveling at 67 ft/sec decelerates at the constant rate of 23 ft/sec2 when thebrakes are applied.(a) Find the velocity v(t) of the car t seconds after the brakes are applied. Then

find its distance s(t) from the point where the brakes are applied.(b) Use the graphing utility of your calculator to sketch the graphs of v(t) and s(t)

on the same screen (use [0, 5]1 by [0, 200]10).(c) Use trace and zoom to determine when the car comes to a complete stop and

how far it travels in that time. How fast is the car traveling when it has trav-eled 45 feet?

Recall (from Section 2.5) that according to the chain rule, the derivative of the func-tion (x2 � 3x � 5)9 is

[(x2 � 3x � 5)9] � 9(x2 � 3x � 5)8(2x � 3)

Notice that the derivative is a product and that one of its factors, 2x � 3, is the deriv-ative of an expression, x2 � 3x � 5, which occurs in the other factor. More precisely,the derivative is a product of the form

g(u)

where, in this case, g(u) � 9u8 and u � x2 � 3x � 5.

You can integrate many products of the form g(u) by applying the chain rule

in reverse. Specifically, if G is an antiderivative of g, then

since, by the chain rule,

To summarize:

d

dx[G(u)] � G�(u)

du

dx� g(u)

du

dx

�g(u)du

dx dx � G(u) � C

du

dx

du

dx

d

dx


Integration bySubstitution

2

Integration - Lamar University calculus/hoffman/ch05sec01.pdfx, and C is called the constant of...

Documents

Transcript of Integration - Lamar University calculus/hoffman/ch05sec01.pdfx, and C is called the constant of...