Antidifferentiation
TS: Making decisions after
reflection and review
Objectives
 To define antidifferentiation.
 To investigate antiderivatives, indefinite
integrals, and all of their parts.
 To use basic integration rules to find
antiderivatives.
CALCULUS
$200
Its derivative is
2x
What is
f ( x)  x 2 ?
$400
Its derivative is
2
3x
What is
f ( x)  x3 ?
$600
Its derivative is
4x
What is
f ( x)  2 x 2 ?
$800
Its derivative is
1
2
x
What is
1
f ( x)   ?
x
$1000
Its derivative is
x
What is
f ( x)  x 2 ?
2
3
3
Antidifferentiation
 Up to this point in calculus, you have been
concerned primarily with this problem:
given a function, find its derivative
 Many important applications of calculus
involve the inverse problem: given the
derivative of a function, find the function
Antidifferentiation
 This operation of determining the original
function from its derivative is the inverse
operation of differentiation and is called
antidifferentiation.
 Antidifferentiation is a process or operation
that reverses differentiation.
Antidifferentiation
7
What is the antiderivative of F  x   x ?
G  x   18 x 71
 18 x8
Notice:
G  x   F  x 
G is an antiderivative of F.
Antiderivatives & Indefinite Integrals
 The antidifferentiation process is also called integration.
Integral
sign
Differential
(variable of
integration)
 f ( x) dx  F ( x)
Indefinite
Integral
Integrand
Antiderivative
The derivative of F is f.
F x  f  x 
Antiderivatives & Indefinite Integrals
x N 1 , N  1
N
N

1
 x dx 
ln x , N  1
The Power Rule for Integration
This absolute value
prevents you from
having to find the
natural log of a
negative number.
1
When N  1, x  x  1x
N
Q: What function has the derivative 1 ?
x
A: ln x
Antiderivatives & Indefinite Integrals
2
2
x
dx

x

What if we were to shift the graph up 1 unit?
Antiderivatives & Indefinite Integrals
Do the slopes change?
Antiderivatives & Indefinite Integrals
2
2
x
dx

x
1

The slopes stay the same.
Antiderivatives & Indefinite Integrals
 2 x dx  x
2
C
If a function has an antiderivative,
then it has an infinite number of antiderivates.
Antiderivatives & Indefinite Integrals
 2 x dx  x
2
C
Constant of
Integration
To capture the fact that there are infinitely
many antiderivatives we add a constant.
Basic Integration Rules
 (Number) dx
 (Number) x  C
Evaluate  2 dx 
2x  C
Basic Integration Rules
Constant Rule for Integration
 c dx  cx  C
Evaluate  5 dx 
5x  C
Basic Integration Rules
 (Number)  f  x  dx
 (Number)  f  x  dx
The integral of a function times a
constant is equal to the constant
times the integral of the function.
Basic Integration Rules
Evaluate  5 x dx 
3
5 x dx 
3
5
5
4
x4
4
C 
x4  C
Q: How do you know if you have found the correct antiderivative?
A: Take the derivative of your answer to check.
Basic Integration Rules
Constant Multiple Rule for Integration
 c  f  x  dx  c  f  x  dx
Sum & Difference Rules for Integration
  f  x   g  x  dx   f  x  dx   g  x  dx
  f  x   g  x  dx   f  x  dx   g  x  dx
Basic Integration Rules
Evaluate   6 x  4 x  1x +1 dx 
2
1
6
x
dx

4
x
dx



 x dx  1 dx 
2
6
x3
3
4
 ln x  x C 
x2
2
2x 2x  ln x  x  C
3
2
You Try these three.
Evaluate  3t dt 
3
t
3
2
3 t dt  3  c  t  c
3
2
Evaluate  2x dx 
2
2 x dx 
2
-2
x -1
C
2
c 
x
1
Evaluate  5du  5u  c
Evaluate  x(3x  4)dx 
 (3x  4 x)dx 
 3x dx   4xdx 
2
2
x  2x  c
3
2
Conclusion
 Antidifferentiation is a process or operation
that reverses differentiation.
 The antidifferentiation process is also
called integration.
 Similar to differentiation, integration has a
variety of rules that we must remember,
recall, and be able to use.