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.

~ 152. 153. Differentiation when there are infinite values.

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The quantity on the right side of this equation, as was just shown,
converges with h to zero and therefore we have:
Lim rf(x, + h) - f(x, a) dx- S f(x, a) dx.
h=O0 h aJ
a a
b) If either or both of the functions f(x, a) and af become
infinite at the point x = c within the interval, but in such a manner
that integration is permitted for each, while a varies within an interval
a - h to a + Ah, then it is once more a sufficient condition for the
validity of the Theorem of differentiation under the integral sign,
that f shall become determinately infinite in an order lower than
the first but otherwise shall remain continuous. For we have:
c c - d c
a-r' x /^/ - 7 + h) 7 fJ (x,. z.
a I fi (x, a) dx = Li df(X c+ +)-f(imx, I (X a + 7)- f(X' a) dx.
a a c-d
The first of these limiting values passes over, however small 6 may be,
into the value of the integral:
c-a~
a f x(, NIX
the second takes the form:
J a'(x, A + h) d
c-o a
and can by hypothesis, however small h is chosen, exclusively by choice
of 6 be made smaller than an arbitrarily small number. Therefore:
c
aJf(a, a) dx
a
differs arbitrarily little from:
c —
af(, a)
ao dx
as 6 converges to zero; establishing the equation we desired to prove:
C C
(xd, -,a) (xX) d x.
153. Integration of a definite integral with respect to
a parameter.
If the definite integral is a continuous function of the parameter