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.

264 The definite integral as the limiting value of a sun. Bk. II. ch. Vl.

As 7; increases, these integrals converge in amount to zero; hence
the infinite series has for x = o a finite value. It is worthy of
notice further that:
Jsin x 1 sin as (
— X dx =f sf I-, (x == a, a > 0)
j- x z - l '
0 0
so that the value of the integral is independent of a; it is ~t (~ 155).
The convergence of these integrals is only conditional, i. e, arises
only by the changes in sign of the functions to he integrated; the
integrals formed of the absolute amounts are divergent. It has however
been proved by Du Bois-Reymond in a general class of examples
(Math. Ann., Vol. XIII) that integrals can be convergent even in case
of oscillations between zero and positive limits.
151. Differentiation of a definite integral with respect
to a parameter.*)
A theorem stating how the definite integral in certain cases
can be differentiated with respect to a quantity that is contained
in the function to be integrated, is of importance in calculation with
definite integrals.
First of all we remark: When f(x, a) is a continuous function of
both variables x and a (~ 52) within the domain that is determined
by the values x =a to x = b, and a = 3 to a = y, the definite
i nit e g r a 1:
integral:
b
f (x, a) x
is also a continuous function of a. For if, whatever be the value
of x, a value h can be assigned such that in the interval a to a -+ h:
abs [f(x, a + h) -f'(x, a)] < 4,
then will also the absolute value of:
b b b
(X~, a h- ) (I lr; -, a)dx -JL X
a/;(x a X 7X cx ~ l/ c)xi [' a +zbfx i) -Af(x a)] dX
a a a
be smaller than 6(b a); therefore it can be arbitrarily diminished
by choice of h. This is a sufficient condition; but the theorem cannot
be converted.
The differential quotient of the definite integral for a determinate
value of a is to be calculated as a limiting value:
b b
Lr, a +..d.x - ffx, a.)d (..
f l(xLira /'(x, a + h) - f x, ) x;
Lim= — -- - LimJ _ + )f(x,;
a
') Thomri: Einleitung in die Theorie der bestimmten Integrale. p. 20 etc..