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.

~ 147. 148.

Connexion with the indefinite integral.

257

value on passing from one interval into the next; i. e. it remains
constant throughout the entire interval of integration:
x
F'(x)dx - F(x) 0C.
a
The value of this constant is determined by putting x - a; thence
follows:
x
- 1F() C or: JF(x)dx == F() - F(a).
a
Therefore when the derivate Ji'(x) of a continuous function F(x) is
hnown and is finite and integrable, the value of its integral is always
F(x) - F(a); even though we may alter arbitrarily the value of F'(x)
in infinitely many discrete points.
148. Wlhen the function f(x) becomes infinitely great, either
determinately, or, oscillating between arbitrary limits, as the variable x
approaches a certain value c in the interval from a to b, the sum, whose
limiting value is the definite integral, can assume any value whatever
for any finite partition of the interval, it has therefore no limiting value,
b
and Jf(x)dx as hitherto defined would have no meaning. But if
a
under these circumstances the sum:
c- a, b
Jf(x) dx + f(x) dx
a C +t2
assume a fixed limiting value while ca and a2 independently converge
b
to zero:jf(x)dx is understood to mean this limiting value.
a
This was indicated in ~ 106. Examples in which the function to be
integrated becomes infinite occur in ~ 122 Note and ~ 132.
Now the necessary and sufficient condition that each of the two
integrals may have a determinate value, is that
Cf- 1 C 0+ a2
shall vanish, when E is always smaller than a, and a converges to zero.
In case the function becomes determinately infinite at a point,
this condition is certainly fulfilled, when in t h e nei g h b o u r h o o d
of this point it becomes algebraically infinite in lower than
the first order; taking as unity the order of - for x = 0.
TIARNACI, Calculus. 17