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.

260 The definite integral as the limiting value of a sum. Bk. II. eh. V1.
c-a
jf(x)P(x)dx
is smaller than:
c-d
MJ (f p(x) dx,
a
where M signifies the greatest amount 4'(x) assumes in this interval.
But by hypothesis the second factor remains finite even for 6 = 0.
Scholium: When the integrable functions become infinite
always in a determinate manner, their product is integrable; for
then all the hypotheses of the theorem are fulfilled.
But when infinity points coincide, we cannot immediately conclude
anything concerning integrability. Thus, from the integrability of a
function that becomes infinite, we cannot conclude that of the square
of this function: Ex. gr. we have the value of
1 1
i dx d
J / - 2~ -- ' but that of J
0 0
becomes infinite.
The extension of the First Theorem of the Mean Value to the product of two functions, VII., holds although cp(x) becomes determinately
infinite, provided f(x) remains finite. Here moreover the definite integral
is a continuous function of its upper limit, Theorem VIII.. For, c being
an infinity point, we have:
1C(x)dx Lim (j(x)xLim (F(c —)).
a a 6=o d=o
Now since F is a continuous function of 6, we can choose 6 so that
abs [F(c) - F(c -- 6)]
shall become smaller than an arbitrarily small prescribed number e,
provided there be any limiting value F(c).
The Second Theorem of the Mean Value, X., continues valid even
when the function (p(x) becomes infinite; provided only p (x) and
f(x) ((x) are integrable. The differential quotient of the integral with
respect to its upper limit (~ 147) becomes determinately infinite at
each point at which in continuous increase the function that is to be
integrated becomes determinately infinite; when this is not the case,
the differential quotient of the integral needs not coincide with the
value of the function that is to be integrated; it also can become
indeterminate. We have ex. gr. in conformity with the equation:
dX (2 cos (ex)) (- ( 2x cos (ex) q- sin (e) e,