An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 149. 150. Fundanental theorems in case of infinity points.

261

f (2^ osK") +sin ()x e d x==x2 cos 2os(ea)F(x)-F(a)
j (2 x cos (e a)-oS ( )Cos ( Z)-(et).
a
The differential quotient of the function F(x) for x = 0 has the value:
Lim (F(+( h), Lim (h cos (e))= 0
it -0
while the function to be integrated becomes indeterminately infinite at
this point.
150. As already shown in ~ 106, the definite integral can still
have a finite value, even when its limits become infinite, provided we
oo ib
understandby f(x)dx the limiting value assumed by Jf(x)dx
a C
when b =o. Similarly we define:
b b
f(x)dx = — Lim (x) dx, when a = — o).
~-a~o C~a
By the substitution x -- we can always reduce the investigation
of the negatively infinite limit to that of the positive. Examples of
the existence of such limiting values have also been already given,
see specially ~ 107 and ~ 137.
But now the necessary and sufficient condition that there may
be a determinate limiting value, is: that
w
jI(x) dx
shall be smaller than an arbitrarily small quantity 6, when tt is assumed
sufficiently great and to greater than u.
When, for arbitrarily increasing values of x, the function does
not oscillate infinitely often, this condition is certainly fulfilled if,
for x = oo, f(x) vanish algebraically in an order higher than
the first, taking as unity the order of - for x c o. In
other words, f(x) can be integrated up to x = oc when this limit is
a zero or nullity of f(x) whose nullitude > 1). For, if the absolute
values of f(x) form a decreasing series, such that:
abs f(x) < A
A being an arbitrary finite quantity, and v a number greater than 1:
*) How a definite integral with infinite limits can be considered directly as
P = 00o
the limiting value of a sum dpfp, is shown by Dini: Fondamenti, p. 338 etc..,z I