(^204) Integrals
Theorem 4.13 If two functions y and F have the same derivative overan
interval, then there is a constant c such that
y(x) = F(x) + c
for each x in the interval.
Considerable information is packed into the little formula
(4.14) y

f f (x) dx = F(x) + c
in which F(x) is any one particular function whose derivative with
respect to x is, over some interval, the integrand f (x) of the integral
It tells us that, whatever the value of the constant c may be, F(x) + c
is a function y whose derivative with respect to x is the integrand f(x).
Moreover, it tells us that if y is a function whose derivative is the inte-
grand, then there must be a constant c for which y = F(x) + c. The
full meaning of the assertion (4.14) has been stated, and this is what is
important. Simply because we must converse with our fellow men and
must read scientific writings, we must join our fellow men in learning
some terminology. The constant c in (4.14) is a "constant of integration"
and the poor fellow is sometimes said to be "arbitrary." The integral
is called an "indefinite integral" to distinguish it from other types of
integrals that are sometimes called "definite integrals." This rather
strange terminology will not injure us if we do not allow it to interfere
with our understanding of the meaning of (4.14). The assertion "each
indefinite integral of f is the sum of a particular indefinite integral and
a constant of integration" sounds weird but is true. The "meaning"
of the word "indefinite" can be understood if we realize that when c is a
constant, say 416, F(x) + c is an "indefinite integral" of f(x) just as
the mayor of Chicago is an "indefinite citizen" of Chicago.
In caseF'(x) = f(x), G'(x) = g(x), and a, b are constants, differentia-
tion formulas show that
(4.15) f [af(x) + bg(x)] dx = aF(x) + bG(x) + c
and
(4.16) f[af(x) + bg(x)] dx = off(x) dx + bfg(x) dx.
These formulas tell us that "integrals of sums are sums of integrals" and
that "constants can be moved across integral signs." The formulas do
not provide justification for moving functions across integral signs; other-
wise, we could replace 96 by = in the formula
(4.161) f f (x) dx : f (x) f dx = f (x) (x + c)
and eliminate all of our troubles.