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.

~ 148. 149. Te function to be integrated becomes infinite.

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cos (e + - exA sin (e )
which for every finite value of x is identical with the derivate:
dx X cos (e)
becomes completely indeterminate for x - 0, oscillating between
arbitrarily great positive and negative values as x approaches zero.
Its infinitude is infinite. Nevertheless we have:
o ex +
jf |cos (e ) + -a- e sin (eA); dx == x cos (e) + C,
even for the value x = 0; because:
d, x cos (el )lx =x — = cos (e) -- a cos (eA
converges to zero along with a and,.
Corollary. When the function becomes infinite in infinitely many
discrete points of its interval of integration, we can resolve this interval
into a finite number of finite intervals that contain none of the
infinity points, while these latter are included within intervals whose
sum becomes arbitrarily small. The integral has a meaning, provided
the partial integrals, formed for the intervals containing no infinity
points, converge to fixed values when the limits of these intervals are
brought arbitrarily close to the infinity points.
When the infinity points form a linear set, such a definition is
no longer possible, since there are then finite intervals containing
everywhere infinitely many points of the kind.
149. The Theorems given in ~ 146 are somewhat modified by
the occurrence of infinity points. We confine our investigation to the
assumption of a finite number of such points.
Instead of Theorem V., whose proof essentially required that the
function to be integrated should be finite, we obtain the theorem:
When (p(x) and O(x) are two functions integrable from a to b,
each becoming infinite at certain points c, but without any infinity
point of (p coinciding with one of Ap, the product (p(x). P (x) can
be integrated in the same interval whenever the functions (p2(x) and,P (x) formed of the absolute values of (p and 4 remain integrable.
In proving this theorem we need only consider an interval from
a to c, within which neither function becomes infinite, while the
limit c is an infinity point for one, ex. gr. for (p(x). Evidently then:
the amount of
17*