Calculus: Analytic Geometry and Calculus, with Vectors

4.3 Properties of integrals 227

When problems are being solved, it is always convenient to use the bracket

symbol in the formula

(4.361) F(x) ]a = F(b)- F(a).

This symbol can be read "eff of x bracket a, b." The symbol means

exactly what the formula says it does; to obtain its value, we write the

value of F(x) when x has the upper value b and subtract the value of F(x)

3

when x has the lower value a. For example, x3 2 = 27 - 8 = 19. It is

easy to see that the value of the bracket symbol is unchanged when we

add a constant to the function appearing in it. Thus

F(x) + c]b = [F(b) + c] - [F(a) -4- c] = F(b)- F(a).

Therefore, we can put (4.36) in the form

(4.362) f ab f (x) dx= F(x) +c]a

where c is 0 or any other constant. Since we have assumed that f is
continuous over a 5 x 5 b, it is a consequence of Theorem 4.35 that
F'(x) = f(x) when a 5 x < b. Since each function whose derivative
with respect to x is f(x) must have the form F(x) + c, the result (4.362)
can be put in the following form.

Theorem 4.37 If f is continuous over a 5 x < b and if F'(x) = f(x)
when a < x S b, then

f ab f(x) dx=F(x)]'= F(b) - F(a).

In substantially all applications of this theorem, the notation of indefi-
nite integrals is used. In such cases the following version of Theorem
4.37 gives precisely the information we actually use to evaluate integrals.
Theorem 4.38 The formula

(4.381)

fb

f(x) dx = F(x)]a = F(b) - F(a)

is correct if f is continuous over a S x :-:5 b and

(4.382) ff(x) dx = F(x) + c

when a5x5b.
When we are able to find a useful expression for the F(x) in (4.382),
the integral in (4.381) can be evaluated with remarkable ease. We sim-
ply ignore the limits of integration on the first integral until (4.382) has
been obtained and then, taking c = 0 unless it seems desirable to give c
some other value, insert the bracket symbol to obtain (4.381). For
example,
xa
f2 xzdx
3J_

- -2

2