Antiderivative-and-Indefinite-Integrals.ppt

math 207

1. ANTIDERIVATIVES AND
INDEFINITE INTEGRALS
Dianne Tolarba

2. THE ANTIDERIVATIVEexorxorx 
333
3
1
;
3
1
;
3
1
The reverse operation of finding a derivative is called the
antiderivative. A function Fis an antiderivativeof a function fif
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) = x
2
Find several functions that have the derivative ofx
2
Answer:
Theorem 1:
If a function
has more than
one
antiderivative,
then the
antiderivatives
differ by at
most a
constant.

3. 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: x
2
,
x
2
+ 1,
x
2
+ 3,
x
2
-2,
x
2
+ c (c is any real number)

4. The symbol is called an integral sign, and the function f(x) is
called the integrand. The symbol dxindicates 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 Cis called the constant of integration.
INDEFINITE INTEGRALS 
 CxFdxxf )()( 
dxxf)(
Let f(x) be a function. The family of all functions that are
antiderivatives of f(x) is called the indefinite integraland has
the symbol

5. INDEFINITE INTEGRAL
FORMULAS AND PROPERTIES   












dxxgdxxfdxxgxf
dxxfkdxxfk
Cxdx
x
Cedxe
nC
n
x
dxx
xx
n
n
)()()()(.5
)()(.4
||ln
1
.3
.2
1,
1
.1
1
(power rule)
It is important to note that property 4 states that a constant
factor can be movedacross an integral sign. A variable
factorcannotbe movedacross an integral sign.

6. EXAMPLE 1:
A)
B)
C) 
dte
t
16 
dxx
4
3 
dx2 Cx2 Ce
t
16 CxC
x









5
5
5
3
5
3

7. EXAMPLE 1 (CONTINUE)
D)
 dxdxxdxx 132
25 
 dxxx )132(
25 Cx
xx


















 1
3
3
6
2
36 Cxxx 
36
3
1

8. EXAMPLE 2
A) 

 dxxdxx
43
2
32 

 dxxdxx
43
2
32 









dx
x
x
4
3
2
3
2 C
xx

























3
3
3
5
2
33
5 Cxx 
33
5
5
6 C
x
x 
3
3
5
1
5
6

9. EXAMPLE 2 (CONTINUE)
B) 
dww
5
3
4 Cx
5
8
2
5 
dww
53
4 C
w














5
8
4
5
8

10. EXAMPLE 2 (CONTINUE)
C)  
dxxx8
2 
 xdxdxx 8
2 









dx
x
xx
2
34
8 C
xx










2
8
3
23 Cx
x

2
3
4
3

11. EXAMPLE 2 (CONTINUE)
D)  
 dxxxx 623
23 
 dxxdxdxxdxx 623
23 
 dxxx )3)(2(
2 Cx
xxx


















 6
2
2
3
3
4
234 Cxxx
x
 6
4
23
4