Calculus: Analytic Geometry and Calculus, with Vectors

4.1 Indefinite integrals 203

which (4.11) holds, we represent it by the ingenious symbol in the formula

(4.111) F(x) = ff(x) dx or y = ff(x) dx.

The second equation is read "y equals an integral of eff of x dee x."
We should all know that it can be read "y equals an indefinite integral
of eff of x dee x," or "y equals a function whose derivative with respect
to x is f ' or "y equals an antiderivative with respect to x of f," but
simplicity always prevails and we read what we see and say what is to

be written. The integral sign f is an elongated S, the f (x) is called the
integrand, and the dx tells us that derivatives with respect to x are
involved.t This matter turns out to be so important that we must con-
tinually remember the following definition.

Definition 4.12 The indefinite integral in the formula

(4.121) y = ff(x) dx = O(x)

is (if it exists) a function of x whose derivative is the integrand f(x); in other

words, the formula

(4.122) dx= f(x) _ (x)

and the formula (4.121) are both true or both false.

For an example, let us see what we know and can learn about the

functions y for which the equivalent formulas

(4.123) dx = 2x, y =

J

2x dx

are valid. We may remember that we differentiated x2 and got 2x.

Hence a function y for which the formulas are valid might be x2 but it

does not have to be because y might be x2 + 1 or x2 - 5 or x2 + 416.

It can be proved that a given function y will satisfy the equivalent

formulas (4.123) if and only if there is a constant c such that y = x2 + c.

Thus

(4.124) f 2x dx = x2 + c.

To be precise about this matter, we state the following theorem which

will be proved later in a remark following the proof of Theorem 5.57.

I Perhaps it should be emphasized at once that the dx in the symbol is not a number. If

we resist temptations to jump to the conclusion that the dx and the crossbar on the f and

the integral sign are numbers, we overcome a difficulty that makes some people feel that the

good old symbol should be abandoned in favor of another which provides fewer temptations.