integrals 5. integrals in section 5.3, we saw that the second part of the fundamental theorem of...
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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