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Antiderivatives and Indefinite Integration

# Antiderivatives and Indefinite Integration -...

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Antiderivatives and Antiderivatives and Indefinite Indefinite Integration Integration Section 4-1

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Antiderivatives Antiderivatives Find a function F whose derivative is f(x) = 3x 2 You probably said that F(x) = x 3 because F is called an antiderivative of f. Observe that F 1 (x) = x 3 , F 2 (x) = x 3 + 5, and F 3 (x) = x 3 + 97 are all antiderivatives of f . [ ] 2 3 3 x x dx d = [ ] 2 3 3 x x dx d =
Thm. 4.1 Representation of Thm. 4.1 Representation of Antiderivatives Antiderivatives If F is an antiderivative of f on an interval I , then G is an antiderivative of f on the interval I iff G is of the form G(x) = F(x) + C , for all x in I where C is a constant.

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If f(x) = 2x Then G(x) = x 2 + C is the family of all antiderivatives of f(x)= 2x C is the constant of integration. The family of functions represented by G is the general antiderivative of f , and G(x) = x 2 + C is the general solution of the differential eqn. G ’(x) = 2x.
Notation for Antiderivatives Notation for Antiderivatives When solving a differential eqn. of the form dy = f(x) dx

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Antiderivatives and Indefinite Integration -...

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