Integration
In this chapter, we solve, more or less simultaneously, the following problems: (1) Given a function f(x), find a function F(x) such that $$ F'\left( x \right) = f\left( x \right)$$
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Integ ration
In this chapter, we solve, more or less simultaneously, the following problems: (1) Given a function I(x), find a function F(x) such that
F(x) = I(x).
This is the inverse of differentiation, and is called integration. (2) Given a function I(x) which is ~ 0, give a definition of the area under the curve y = I(x) which does not appeal to geometric intuition. Actually, in this chapter, we give the ideas behind the solutions of our two problems. The techniques which allow us to compute effectively when specific data are given will be postponed to the next chapter. In carrying out (2) we shall follow an idea of Archimedes. It is to approximate the function I by horizontal functions, and the area under I by the sum of little rectangles.
IX, §1. THE INDEFINITE INTEGRAL Let I(x) be a function defined over some interval. Definition. An indefinite integral for
F'(x) = I(x)
I
is a function F such that
for all x in the interval.
S. Lang, A First Course in Calculus © Springer Science+Business Media New York 1986
288
[IX, §1]
INTEGRATION
If G(x) is another indefinite integral of f, then G'(x) the derivative of the difference F - G is 0:
= f(x)
also. Hence
(F - G)'(x) = F'(x) - G'(x) = f(x) - f(x) = O. Consequently, by Corollary 3.3 of Chapter V, there is a constant C such that F(x) = G(x)
+C
for all x in the interval. Example 1. An indefinite integral for cos x would be sin x. But sin x + 5 is also an indefinite integral for cos x. Example 2. log x is an indefinite integral for l /x. So is log x logx - n.
+ 10 or
In the next chapter, we shall develop techniques for finding indefinite integrals. Here, we merely observe that every time we prove a formula for a derivative, it has an analogue for the integral. It is customary to denote an indefinite integral of a function f by or
f
f(x) dx.
In this second notation, the dx is meaningless by itself. It is only the full expression Jf(x) dx which is meaningful. When we study the method of substitution in the next chapter, we shall get further confirmation for the practicality of our notation. We shall now make a table of some indefinite integrals, using the information which we have obtained about derivatives. Let n be an integer, n # - 1. Then we have
f If n =
-
X"+l
x" dx
= n + 1·
1, then
f~
dx = log x.
(This is true only in the interval x > 0.)
[IX, §1]
289
THE INDEFINITE INTEGRAL
In the interval x > 0 we also have
I
xc+l
xCdx = - c+ 1
for any number c # - 1. The following indefinite integrals are valid for all x.
I
I I
sin x dx = - cos
cos x dx = sin x,
I
-_l_2 dx 1+ x
eX dx = eX,
X,
= arctan x.
Finally, for - 1 < x < 1, we have
Ip
1 - x2
dx
= arcsin x.
In practice, one frequently omits mentioning over what interval the various functions we deal with are defined. However, in any specific problem, one has to keep it in mind. For instance, if we write
this is valid for x > 0 and is also valid for x < O. But 0 cannot be in any interval of definition of our function. Thus we could have
f when x < 0 and
I
x-
1/3
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