Antiderivative
A function F(x) is an antiderivative of f(x) on I if F’(x) = f(x).
– There are an infinite number of antiderivatives. Why?
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An antiderivative is a function which has the given function as its derivative.
• Example: f(x) = 2x
It is also called an indefinite integral.
Antidifferentiation, or integration, is the process of finding antiderivatives.
The notation for integration is f ( x)dx F ( x) c .
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•
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is the integral sign.
dx is the differential of x and denotes the variable to integrate with respect to.
c is the constant of integration.
Representation of Antiderivative
Theorem 1
If F(x) is an antiderivative of f(x) on the interval I, then G(x) is an antiderivative of f(x) if and
only if G(x) is of the form G(x) = F(x) + c where c is a constant.
Proof
(<=) Use the definition of an antiderivative.
(=>) Show H(x) = G(x) – F(x) is constant by assuming it is not constant and apply the Mean
Value Theorem.
Integration Rules
0dx
c
cos xdx
kdx
kx c
sin xdx
cos x c
sec2 xdx
tan x c
kf ( x)dx
f ( x)
k f ( x)dx
g ( x)dx
f ( x)dx
n 1
x n dx
x
c, n
n 1
1
g ( x)dx
sin x c
sec x tan xdx
csc 2 xdx
sec x c
cot x c
csc x cot xdx
csc x c
These are direct implications of differentiation.
Just with differentiation, you must put the function in the correct form.
There are no product, quotient, or chain rules.
You will see additional techniques in Calculus II.
Examples
f(x) = 3x2 – 7x + 2
f ( x)
2 x3 3 x
f(x) = 5x3/2 – 2x-2 + cosx
5
x3
csc2 x
f ( x)
3
2x
2
3x5
sec x(sec x tan x)