An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

42

SUMMATION OF DIFFERENCES

The remaining problems are tabulated in order in the three
columns below, the first giving the problem, the second the
function whose derivative is to be of a given form, and the third
the result:
r,.. 4 b4 a4
x3dx, d,Ja 4 4 4
roux I I I
X3 22 2a2b
Ja 3, -2,2' 2 n2 2 b
rb
cosdx sin, sin b - sin a.
obdx
j -x logr, log b - log a.
b
35. The expression r(x)dx is called the definite inSegnal
of f (x), or of (x)dx, betzeen the limits a and b. Now the fundamental theorem tells us that the value of this definite integral can
be found by finding the function whose differential coefficient
is +(x), substituting first b, and then a, for x in this function, and
subtracting the latter result from the former. We then very
properly define j(xr)dr, to mean any function of x, which,
when differentiated, gives 4(x). Thus, let (.zr) be such a
function; then d. (r() + c = -I (xr) -= (.r). Hence there
are an infinite number of such functions, any two of which
differ by a constant, but any one of which satisfies the
conditions of the theorem. We call f (.x)dx the in'zdfinite
integral of +(x); it is a function which, when found, generally
leads to the definite integral between the given limits. The
indefinite integrals in the above problems are -+c, 7vo0+t O2+r,
30 2
I X I
— +~c -,, - + c, sin x + c, and log x + c. Hence we
x 4 2 x"2
see that F(x)ddx is not a function of x. What is it a function of? Is F(x)dx a function of x?