Indefinite Integral

Antiderivatives
We have studied how to find the derivative of a function. However, many problems require
that we recover a function from its known derivative (from its known rate of change).
For instance, we may know the velocity function of an object falling from an initial
height and need to know its height at any time over some period. More generally, we want
to find a function F from its derivative ƒ. If such a function F exists, it is called an antiderivative
of ƒ.
Finding Antiderivatives
4.8
DEFINITION Antiderivative
A function F is an antiderivative of ƒ on an interval I if
for all x in I.
F?sxd = ƒsxd
The process of recovering a function F(x) from its derivative ƒ(x) is called antidifferentiation.
We use capital letters such as F to represent an antiderivative of a function ƒ, G
to represent an antiderivative of g, and so forth.
EXAMPLE 1 Finding Antiderivatives
Find an antiderivative for each of the following functions.
(a)
(b)
(c)
Solution
(a)
(b)
(c)
Each answer can be checked by differentiating. The derivative of is 2x. The
derivative of is cos x and the derivative of is
The function is not the only function whose derivative is 2x. The function
has the same derivative. So does for any constant C. Are there others?
Corollary 2 of the Mean Value Theorem in Section 4.2 gives the answer: Any two antiderivatives
of a function differ by a constant. So the functions where C is an
arbitrary constant, form all the antiderivatives of More generally