Formal Definition of
Antiderivative and Indefinite
Integral
Lesson 5-3
Back in chapter 1 we were given an
outline of the topics of Calculus:
• Limits
• Derivatives
• One Type of Integral
(definite integral)
• the Other Integral
Definite Integral
• The process of evaluating a product in which
one factor varies.
• Graphically, the area under the curve.
Back in lesson 1-4 we learned about
using trapezoids to approximate this
area and applied the trapezoid rule.
The Antiderivative is just another name for the
Indefinite Integral.
It is considered “Indefinite” because of the
“+ C”, which we don’t know until we are given a starting
point relative to our function.
So, when asked to find an equation for the
antiderivative dy  3 x 2dx
We know
3
3x
y
c
3
y  x c
3
What we didn’t know at the time, when finding
antiderivative, was that we were writing the
answers to the indefinite integrals.
The antiderivative of dy  3 x 2dx
Is the same as evaluating
3x
dx

2
integrand
• The purpose of discussing the
differential in the previous lesson was
because there is a relationship between
it and the integral.
• It tells us what variable we are
performing the integration with respect
to.
• It comes in handy when dealing with
those composite functions.
Definition of Indefinite Integration:
g ( x )   f ( x )dx if and only if g '( x )  f ( x )
g ( x)  x
g '( x)  4 x
3

4
f ( x)  4 x
3
4
4x
c
4x dx 
4
3
 x c
4
g ( x)   f ( x)dx
Two properties of Indefinite Integrals:
• Integral of a Constant times a Function:
 kf ( x)dx  k  f ( x)dx
constants can be pulled through the integral sign.
Beware the variable can NOT!
1
4 3
t dt
4

3
1
t 3 dt
3
4
4t 3 
  c t 3 c
3  4 3 
4
Two properties of Indefinite Integrals:
• Integral of a Sum of Two Functions:

f ( x)  g ( x)dx   f ( x)dx   g ( x)dx
integration distributes over addition.
 3sin x  sin 3xdx
 3sin xdx   sin 3xdx  3 sin xdx   sin 3xdx
 cos3x
 3cos x 
c
1
3
 3cos x  cos3 x  c
3
Integrating Composite Functions Using USubstitution:
• Power Rule for Integration:
n≠-1
u
u
du


c

n 1
 ( x  2)

6
u
u du   c
6
5
n 1
n
5
dx
Let u = x+2
1
6
 ( x  2)  c
6
Integrating Composite Functions Using USubstitution:


4 x  1dx
3
2
Let u  4 x  1
 (4 x  1) 2  c
3
3
2
1
3
2
u
2
2

(
u
)
c


c
u du 3
3
2
Integrating Composite Functions Using USubstitution:
Let u  4 x  1
Since u  4 x 1
u '  4dx
1
So u  dx
4


4 x  1dx
3
2
 (4 x  1) 2  c
3
1
4 x  1dx   u du
4
1
2
3
1 u2
 
c
4 32
Integrating Composite Functions Using USubstitution:

4 x  1dx
3
2
1 12
1 2 32
1
u
  u du  
 c   (u )  c
4
4 3
4 32
1 32
 (u )  c  1 (4 x  1) 3 2  c
6
6
3
2
 (4 x  1) 2  c
3
1
3

(4 x  1)  c
6
Property:
If f ( x)  e and f '( x)  e
then
x
x
e
dx

e

c

x
x
Integration of e
x
Example : Evaluate the indefinite integral
6 x

4
e

 4 e
6 x
dx
dx
2 6 x
 e c
3
let u  6 x
u '  6dx
1 
 4 eu du
6
4 u
e c
6
Back in lesson 3-9 we learned to differentiate the exponential
function 2x. With the following definition we can also
integrate this exponential with a base other than e.
Definition: Exponential with Base b
b e
x
2 e
x
x ln b
x ln 2
x
x ln 2
2
dx

e

 dx
let u  x ln 2
du  ln 2dx
1
1 u
u

e du 
e c

ln 2
ln 2
x ln 2
x
e
2

c 
c
ln 2
ln 2
Properties: Derivative and Integral of an
Exponential Function
d x
x
b

b
ln b


dx
1
 b dx  b ln b  c
x
x
Ω