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.

262 The definite integral as the limiting value of a sum. Bk. TII. cb. VI.

ww
j'(x) dx will be smaller than A -- (w~_v - tl< U
and while 1 - v < 0 this expression converges to zero as the values
of u and w increase.
But the condition stated is not fulfilled when the function vanishes
in the first or a lower order, or when it remains finite. For, if
A
abs f(x) > -, (< 1)
x
we shall have
Jit(x)dx> AJ 'dx - - (t -v (, - u
Ub tU
and here the exponents of the arbitrarily ncrincreasing values u and w
are positive.
On the whole the investigation is evidently quite analogous to that
in 148, because by the substitution x -= -, j f(x)dx passes over into
-Jf(z);2
accordingly the behaviour of the new integral at the point =- 0
must be examined.
Our criterion shows for instance, without any substitution, that
the integral (~ 137):
J X
must have a finite value; for, the function:
becomes smaller than
x XV
for every v > 1, because:
Lim (x1-1 ~ c-x) = 0.
But when the function makes infinitely many oscillations as the
values of x become infinite, its nullitude needs no restriction in order
that the condition may be fulfilled. Thus ex. gr. the value of:
oo
Jsin (x2)dx
is finite and determinate, although for x-==oo the function to be integrated
becomes quite indeterminate between the limits -1 and +- l.) For;

*) Dirichlet, Journal f. Math., Vol. 17.