2. Objectives
To define antidifferentiation.
To investigate antiderivatives, indefinite
integrals, and all of their parts.
To use basic integration rules to find
antiderivatives.
9. 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
10. 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.
11. Antidifferentiation
7
F x x
G x
8
1
8 x
7 1
x
1
8
What is the antiderivative of ?
Notice:
G x F x
G is an antiderivative of F.
12. Antiderivatives & Indefinite Integrals
( )
f x dx
Differential
(variable of
integration)
Integral
sign
Integrand
Antiderivative
The antidifferentiation process is also called integration.
Indefinite
Integral
( )
F x
The derivative of F is f.
F x f x
13. Antiderivatives & Indefinite Integrals
N
x dx
1
N
x
1
N
, 1
N
1
N
1 1
When 1, N
x
N x x
Q: What function has the derivative ?
1
x
A: ln x
ln ,
x
This absolute value
prevents you from
having to find the
natural log of a
negative number.
The Power Rule for Integration
17. Antiderivatives & Indefinite Integrals
If a function has an antiderivative,
then it has an infinite number of antiderivates.
2
2x dx x C
18. Antiderivatives & Indefinite Integrals
2
2x dx x C
To capture the fact that there are infinitely
many antiderivatives we add a constant.
Constant of
Integration
21. 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.
22. Basic Integration Rules
3
Evaluate 5x dx
3
5 x dx
5
4
4
x
C
4
5
4 x C
Q: How do you know if you have found the correct antiderivative?
A: Take the derivative of your answer to check.
23. Basic Integration Rules
c f x dx c f x dx
Constant Multiple Rule for Integration
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
24. Basic Integration Rules
2 1
Evaluate 6 4 +1
x
x x dx
2 1
6 4 1
x
x dx x dx dx dx
6
3
3
x
C
3
2x
4
2
2
x
ln x
x
2
2x
ln x x C
25. You Try these three.
2
Evaluate 2x dx
Evaluate 5du
2
3 t dt
2
Evaluate 3t dt
Evaluate (3 4)
x x dx
3
t
3
3
c
3
t c
5u c
2
2 x dx
-1
x
2
1
c
-2
x
C
2
(3 4 )
x x dx
2
3 4
x dx xdx
3 2
x 2x c
26. 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.