INTEGRALSINTEGRALS

5

INTEGRALS

In Section 5.3, we saw that the second part of

the Fundamental Theorem of Calculus (FTC)

provides a very powerful method for

evaluating the definite integral of a function.

This is assuming that we can find an antiderivative of the function.

5.4Indefinite Integrals and

the Net Change Theorem

In this section, we will learn about:

Indefinite integrals and their applications.

INTEGRALS

INDEFINITE INTEGRALS AND NET CHANGE THEOREM

In this section, we:

Introduce a notation for antiderivatives.

Review the formulas for antiderivatives.

Use the formulas to evaluate definite integrals.

Reformulate the second part of the FTC (FTC2) in a way that makes it easier to apply to science and engineering problems.

INDEFINITE INTEGRALS

Both parts of the FTC establish

connections between antiderivatives

and definite integrals.

Part 1 says that if, f is continuous, then is an antiderivative of f.

Part 2 says that can be found by evaluating F(b) – F(a), where F is an antiderivative of f.

( )x

af t dt

( )b

af x dx

INDEFINITE INTEGRALS

We need a convenient notation for

antiderivatives that makes them easy

to work with.

Due to the relation given by the FTC between

antiderivatives and integrals, the notation

∫ f(x) dx is traditionally used for an

antiderivative of f and is called an indefinite

integral.

Thus, ∫ f(x) dx = F(x) means F’(x) = f(x)

INDEFINITE INTEGRAL

INDEFINITE INTEGRALS

For example, we can write

Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).

3 32 2because

3 3

x d xx dx C C x

dx

INDEFINITE VS. DEFINITE INTEGRALS

You should distinguish carefully between

definite and indefinite integrals.

A definite integral is a number.

An indefinite integral ∫ f(x) dx is a function (or family of functions).

( )b

af x dx

The connection between them is given

by the FTC2.

If f is continuous on [a, b], then

( ) ( )bb

a af x dx f x dx

INDEFINITE VS. DEFINITE INTEGRALS

INDEFINITE INTEGRALS

The effectiveness of the FTC depends

on having a supply of antiderivatives

of functions.

Therefore, we restate the Table of Antidifferentiation Formulas from Section 4.9, together with a few others, in the notation of indefinite integrals.

INDEFINITE INTEGRALS

Any formula can be verified by differentiating

the function on the right side and obtaining

the integrand.

For instance, 2

2

sec tan

because

(tan ) sec

x dx x C

dx C x

dx

TABLE OF INDEFINITE INTEGRALS

1

2 2

( ) ( ) [ ( ) ( )]

( ) ( )

( 1)1

sin cos cos sin

sec tan csc cot

sec tan sec csc cot csc

nn

cf x dx c f x dx f x g x dx

f x dx g x dx

xk dx kx C x dx C n

n

x dx x C x dx x C

x dx x C x dx x C

x x dx x C x x dx x C

Table 1

INDEFINITE INTEGRALS

Recall from Theorem 1 in Section 4.9 that

the most general antiderivative on a given

interval is obtained by adding a constant to

a particular antiderivative.

We adopt the convention that, when a formula for a general indefinite integral is given, it is valid only on an interval.

INDEFINITE INTEGRALS

Thus, we write

with the understanding that it is valid on

the interval (0, ∞) or on the interval (-∞, 0).

2

1 1dx C

xx

INDEFINITE INTEGRALS

This is true despite the fact that the general

antiderivative of the function f(x) = 1/x2,

x ≠ 0, is:

1

2

1if 0

( )1

if 0

C xxF x

C xx

INDEFINITE INTEGRALS

Find the general indefinite integral

∫ (10x4 – 2 sec2x) dx

Using our convention and Table 1, we have:

∫(10x4 – 2 sec2x) dx = 10 ∫ x4 dx – 2 ∫ sec2x dx = 10(x5/5) – 2 tan x + C = 2x5 – 2 tan x + C

You should check this answer by differentiating it.

Example 1

INDEFINITE INTEGRALS

Evaluate

This indefinite integral isn’t immediately apparent in Table 1.

So, we use trigonometric identities to rewrite the function before integrating:

Example 2

2

cos 1 cos

sin sinsin

csc cot csc

d d

d C

2

cos

sind

INDEFINITE INTEGRALS

Evaluate

Using FTC2 and Table 1, we have:

Compare this with Example 2 b in Section 5.2

Example 33 3

0( 6 )x x dx

34 23 3

00

4 2 4 21 14 4

814

( 6 ) 64 2

3 3 3 0 3 0

27 0 0 6.75

x xx x dx

INDEFINITE INTEGRALS

Find

Example 4

12

0( 12sin ) x x dx

The FTC gives:

This is the exact value of the integral.

INDEFINITE INTEGRALS Example 4

212 12

00

212

12sin |12( cos )2

(12) 12(cos12 cos0)

72 12cos12 12

60 12cos12

x

x x dx x

INDEFINITE INTEGRALS

If a decimal approximation is desired, we can

use a calculator to approximate cos 12.

Doing so, we get:

12

012sin 70.1262 x x dx

Example 4

The figure shows the graph of the integrand

in the example.

We know from Section 5.2 that the value of the integral can be interpreted as the sum of the areas labeled with a plus sign minus the area labeled with a minus sign.

INDEFINITE INTEGRALS

INDEFINITE INTEGRALS

Evaluate

First, we need to write the integrand in a simpler form by carrying out the division:

Example 52 29

21

2 1t t tdt

t

2 29 9 1 2 2

21 1

2 1(2 )

t t tdt t t dt

t

Then,9 1 2 2

1

93 2 1

32 1

93 2

1

3 2 3 22 1 2 13 9 3 1

1 2 49 3 9

(2 )

21

2 12

3

(2 9 9 ) (2 1 1 )

18 18 2 1 32

t t dt

t tt

t tt

INDEFINITE INTEGRALS Example 5