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.

~ 105-107. Extension to discontinuous functions. 181
106. Hitherto the formula
x
F(x) =ff(x) dx + Const.
a
has been derived in the sense of expressing the indefinite integral.F(x) from the definite integral; it therefore requires the calculation
of the limiting value of a sum with arbitrarily many summands. It
can however inversely be employed in calculating this limiting value
of a sum, when the indefinite integral 1'(x) is known. But now since in
the differential calculus the derived functions 1(x) = F'(x) belonging
to whole classes of functions F(x) have been calculated; inversely, the
indefinite integral F(x) belonging to each of these, derived functions
f(x) is also known. On the hypothesis that this integral is
unique and continuous in the interval from a to b, we obtain
by the difference of the values of this function the definite integral
from a to b. This calculation is still valid when in the interval
from a to b the function f(x) assumes infinite values, while F(x)
remains finite, since at such a point c we have to put:
Jf(x) dx = Lim i f(x) dx = Lim { F(c - ) F(6 ) } = F(c) - F(a),
a a for a= 0 for =o0
and it holds even for an infinite interval from a to co, or from
- oo to + co, when the function F(x) retains a determinate finite
value actually at these limits, if we introduce the definition:
+-00 w.f (x) dx - Lim (x) dx Lim { F(w) - F(a)} — = F(cxo) F(a).
Ca a w=oo w= —
The indefinite integral of f(x) is usually denoted by the symbol
dff(x) xoo x
ff(x) dx + const., so that d(x = f(x); f(x) is called the function
that is to be integrated; F(x) the integral function.
107. Fundamental formulas:
dx def m
1) + ~ _ x md, (m} -1).
2) cd(lx) = - 3) d em xdx.
dsinzmx d cos nx d zcmx
4) s cos n x 5dx. 5) -sin mxdxo
dx dx
6) d tanx-' 7) -d ccot x (s
dx dx
8) d sin-1 x — x 9) -d cos-'x --
8Y) 1-' 1 V/1-x2

dx10)
10) dtan-ix
1 + x2

11) dcot —X
11) — dcot-lx
I + X2