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Section 3.9 Antiderivatives

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1 Section 3.9 Antiderivatives
Math 1231: Single-Variable Calculus

2 Antiderivative: Definition
Definition A function F(x) is called an antiderivative of f on an interval I if F ‘ (x) = f(x) for all x in I. f(x) antiderivative F(x) derivative

3 General antiderivative
Theorem If F is an antiderivative of f on an interval I, then the general antiderivative of f on I is F(x) + C, where C is an arbitrary constant. Any two antiderivatives of f(x) only differ by a constant.

4 Antiderivative Table Function Particular antiderivative
General antiderivative f(x) F(x) F(x) + C f(x) + g(x) F(x) + G(x) F(x) + G(x) + C c*f(x) c*F(x) c*F(x) + C xn (n≠1) xn+1/(n+1) xn+1/(n+1) + C sin(x) - cos(x) - cos(x) + C cos(x) sin(x) + C sec2(x) tan(x) tan(x) + C sec(x)tan(x) sec(x) sec(x) + C

5 Examples Example Find the general antiderivative of the function f(x) = 3cos(t) – 4sin(t). Example Find the general antiderivative of the function f(x) = (1+ t + t2)/sqrt(t). Example Find f given f ’’ (x) = (2/3) x2/3. Example Find f given f ’ (x) = x*sqrt(x) and f(1) = 2.


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