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.