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5.1:
Antiderivatives and Indefinite Integrals
Introduction to Antiderivative
If f
’
(x) = x², then f(x) = _________________?
If
F
(x) represents the “original” function, then we say
F
(x) =
?³
3
is an
antiderivative
of the function
f(x) = x² because
F’
(x) =
𝑑
𝑑?
(
?
3
3
)
=
x²
However, there are
many
functions that have a derivative of f(x) = x², such as:
In summary,
antidifferentiation
of a given function results in ________________________.
In our example, we would write: ______________________________________________.
The Indefinite Integral
∫
?(
?)??
∫ ?(?)??
=
F(x) +
C
if F
’
(x) = f(x)
integral sign
integrand
constant of integration
Antiderivative
A function
F
is an
antiderivative
of a function
f
if:
𝑭
′
(𝒙) = 𝒇(𝒙)
Objectives
▪
The student will be able to formulate problems involving antiderivatives.
▪
The student will be able to use the formulas and properties of antiderivatives and
indefinite integrals.
▪
The student will be able to solve applications using antiderivatives and indefinite
integrals.

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Example:
Find each indefinite integral
a)
∫ 5??
b)
∫ 9?
?