1. §4.9 Antiderivatives There are two branches in calculus:
Differential calculus
Integral calculus
Math 110 (Differential Calculus):
Given f(x), find f (x)
Applications of derivatives
Math 116 (Integral Calculus):
Given f(x), find F(x) such that F (x)=f(x), F is called an antiderivative of f
Applications of antiderivatives
Math 116 is the inverse of Math 110

2. Def: A function F is called an antiderivative of f on an interval I if F (x) = f(x) for any xI
Th: If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is
F(x) + C,
where C is an arbitrary constant
Note: There are infinitely many antiderivatives for a given f. The difference between them is a constant.

4. Th: Suppose:
(i) f(x)  0 for any x[a,b]
(ii) f is continuous on [a,b]
then: F(x) (representing the shaded area below and viewed as a function of x ) is an antiderivative of f(x) on [a,b]

5. Table of antidifferentiation formulas: