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Antidifferentiation 4.1 OBJECTIVE
• Find an antiderivative of a function.
• Evaluate indefinite integrals using the basic integration formulas.
• Use initial conditions, or boundary conditions, to determine an antiderivative.
2012 Pearson Education, Inc.
All rights reserved Slide 4.1-2
4.1Antidifferentiation
THEOREM 1
The antiderivative of f ( x) is the set of functions
F x +C such that
dx�(x) +C = f (x).
The constant C is called the constant of integration.
2012 Pearson Education, Inc. All rights reserved Slide 4.1-3
4.1Antidifferentiation
Integrals and Integration Antidifferentiating is often called integration.
To indicate the antiderivative of x2 is x3/3 +C, we write òx2dx = x3 +C, where the notation ò f (x)dx
is used to represent the antiderivative of f (x).
More generally, ò f (x)dx = F(x)+C, where
F(x) + C is the general form of the antiderivative of f (x).
2012 Pearson Education, Inc. All rights reserved Slide 4.1-4
4.1Antidifferentiation
Example 1: Determine these indefinite integrals. That is, find the antiderivative of each integrand:
a.) 8dx =8x+C Check: dx(8x+C) =8
b.) 3x2dx = x3 +C Check:
d
dx
x3 +C) =3x2
c.) exdx = ex +C Check:
d x
dx
+C) = ex
d.) 1dx = lnx+C Check: dx(lnx+C) =
1
x
2012 Pearson Education, Inc. All rights reserved Slide 4.1-5
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