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.

Second Chapter.
The integral of rational algebraic functions. Partial fractions.*)
109. The integral of the rational integer function of the nth order:
f(x) == a + ax + a2x2 -. *+ -acx
is, by Theorems a) and b) ~ 108 and by Fundamental Formula 1) ~ 107:
F(x) =ff(x)d c x - aoJd f i~ aIxdx + + oj, xn dx
X" X- xn+1
=Cx + a + a * + n -_ n ~ + Const.
110. A rational fractional function: f(x)- = (x), p being of the
qp (x)
mnth and p of the nth order, when mn n can always be resolved
into an integer function and a proper fractional function, i. e. a
function in which the order of the numerator is at most n - 1.
This only requires the division by the denominator (p(x) to be
carried out until the order of the remainder becomes less than n. As
the integration of the integer function has been given already, we
have only to determine the integral of the form:
f (x) dx
in which 4' is of lower order than (p and they have no common root.
This proper fractional function can be resolved into a sum of
fractions with constant numerators and with denominators that are
linear functions or powers of linear functions.
Partial fractions.
111. Let al, a,...o the n roots or vanishing points of
() (X) == a0 + a — a1 + * + Xn
be real or complex, but first let them all be different. The coefficient
of the highest power of x in (p is supposed unity, it can always be
made so by putting the original factor before the entire quotient
*) Leibnitz and John Bernoulli: Acta erud., 1702 —1703. Euler: Institutiones
calculi integralis. Vol. I. Sect. I. Cap. I.