Post on 17-Jan-2016
13.1 Antiderivatives and Indefinite Integrals
The AntiderivativeThe reverse operation of finding a derivative is called the
antiderivative. A function F is an antiderivative of a function f if
F ’(x) = f (x).
1) Find the antiderivative of f(x) = 5
Find several functions that have the derivative of 5
Answer: 5x; 5x+ 1; 5x -3;
2) Find the antiderivative of f(x) = x2
Find several functions that have the derivative of x2
Answer: exorxorx 333
3
1;
3
1;
3
1
Theorem 1:
If a function has more than one antiderivative, then the antiderivatives differ by at most a constant.
• The graphs of antiderivatives are vertical translations of each other.
• For example: f(x) = 2x
Find several functions that are the antiderivatives
for f(x)
Answer: x2,
x2 + 1,
x2 + 3,
x2 - 2,
x2 + c (c is any real number)
The symbol is called an integral sign, and the function f (x) is called the integrand. The symbol dx indicates that anti-differentiation is performed with respect to the variable x.By the previous theorem, if F(x) is any antiderivative of f, then
The arbitrary constant C is called the constant of integration.
Indefinite Integrals
CxFdxxf )()(
Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol dxxf )(
Indefinite Integral Formulas and Properties
dxxgdxxfdxxgxf
dxxfkdxxfk
Cxdxx
Cedxe
nCn
xdxx
xx
nn
)()()()(.5
)()(.4
||ln1
.3
.2
1,1
.11
(power rule)
It is important to note that property 4 states that a constant factor can be moved across an integral sign. A variable factor cannot be moved across an integral sign.
Example 1:
A)
B)
C)
dtet16
dxx43
dx2 Cx 2
Cet 16
CxCx
55
5
3
53
Example 1 (continue)
D)
dxdxxdxx 132 25
dxxx )132( 25
Cxxx
1
33
62
36
Cxxx 36
3
1
Example 1 (continue)
E)
dxedxx
x45
dxe
xx4
5
Cex x 4ln5
dxedxx
x41
5
Example 2
A)
dxxdxx 43
2
32
dx
xx
43
2 32
Cxx
33
35
233
5
dxxdxx 43
2
32
Cxx 33
5
5
6
Cx
x 3
3
5 1
5
6
Example 2 (continue)
B)
dww5
3
4 dww5 34
Cw
58
45
8
Cx 5
8
2
5
Example 2 (continue)
C)
xdxdxx 82
dx
x
xx2
34 8
Cxx
28
3
23
dxxx 82
Cxx
23
43
Example 2 (continue)
D)
dxxdxx 2
1
3
1
68
dxx
x6
83
Cxx
21
6
34
82
1
3
4
Cxx 126 3
4
Example 2 (continue)
E)
dxxdxdxxdxx 623 23
dxxx )3)(2( 2
Cxxxx
6
22
33
4
234
dxxxx 623 23
Cxxxx
64
234
Example 3Find the equation of the curve that passes through (2,6) if its
slope is given by dy/dx = 3x2 at any point x.
The curve that has the derivative of 3x2 is
Since we know that the curve passes through (2, 6), we can find out C
dxx23
Cxy 3
CxCx
33
33
C 326
C862C
Therefore, the equation is y = x3 - 2
Example 4Find the revenue function R(x) when the marginal revenue is
R’(x) = 400 - .4x and no revenue results at a 0 production
level. What is the revenue at a production of 1000 units?
The marginal revenue is the derivative of the function so to find the
revenue function, we need to find the antiderivative of that function
So R(x) = 400x -.2x2, we know need to find R(1000)
dxx)4.400( 22
2.4002
4.400 xxCx
x
000,200)1000(2.)1000(400)1000( 2 R
Therefore, the revenue at a production level of 1000 units is $200,000
xdxdx 4.400
Example 5The current monthly circulation of the magazine is 640,000 copies. Due
to the competition from a new magazine, the monthly circulation is
expected to decrease at a rate of C’(t)= -6000t1/3 copies per month, t is
the # of months. How long will it take the circulation of the magazine to
decrease to 460,000 copies per month?
We must solve this equation: C(t) = 460,000 with C(0) = 640,000
To find the function C(t), take the antiderivative
dtt 3
1
6000 CtCt
3
43
4
4500
34
6000
C
C
CC
000,640
04500000,640
04500)0(
3
4
3
4
000,6404500)( 3
4
ttC
t
t
t
t
t
9.15
40
40
4500000,180
000,6404500000,460
4
3
3
4
3
4
3
4
3
4
4
3
So, it takes about 16 months