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3.9 Antiderivatives

3.9 Antiderivatives - The operation of finding all...

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Antiderivatives Section 3.9

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Antiderivatives Find a function F whose derivative is f(x) = 3x2 You probably said that F(x) = x3 because F is called an antiderivative of f. Observe that F1(x) = x3, F2(x) = x3 + 5, and F3(x) = x3+ 97 are all antiderivatives of f . [ ] 2 3 3 x x dx d = [ ] 2 3 3 x x dx d =
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If f(x) = 2x Then G(x) = x2 + 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) = x2 + C is the general solution of the differential eqn. G ’(x) = 2x.
Notation for Antiderivatives When solving a differential eqn. of the form dy = f(x) dx write it in the equivalent form dy = f(x)dx.

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Unformatted text preview: The operation of finding all solutions of this eqn. is called antidifferentiation or indefinite integration . integrand variable of integration = F(x) + C constant of integration ∫ ∫ ∫ = = = dx x f y dx x f dy dx x f dy ) ( ) ( ) ( Basic Integration Rules Integration is the “inverse” of differentiation Differentiation is the “inverse” of integration ∫ + = ′ C x F dx x F ) ( ) ( [ ] ) ( ) ( x f dx x f dx d = ∫ Joke Time What kind of snake is good at math? An adder There were 7 copycats in a box. One jumped out. How many were left? None What is big, gray and wears glass slippers? Cinderellaphant!...
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3.9 Antiderivatives - The operation of finding all...

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