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.

200 The integral of explicitly irrational functions. Bk. III. ch. 11I.
'(x, -/Z) G (x) + -1,(x) V/R
' G1 (x) + H1 (x) /R '
G(, H Gl, HL signifyiing integer rational functions. If we rationalise
the denominator of this quotient by multiplying by
G,(x) - H(X)) y,
we have:
F(x. J/iR) == (x) + P (x) J,
9 and 4 denoting rational functions. We are concerned henceforth
only with the irrational part, which may be presented in the form:
X. R f(x) 0 XAn ( Bn 1_
V1 IV~? n-0VR dJ ( xC -~n Vli
for, the rational function f'(x) breaks up into an integer function
and a proper fraction that can be resolved into partial fractions.
The integral Jf -- is reduced by means of a recurring formula
to algebraic functions and to the integral | - In fact:
d (Xn-l l-Va - 2bx + cx2)
= -. i x'f/a -j- 2bx -\ C + 1/+ +-c) V + VX
('n-':-2( + 2~+~x~) — I )
_ __T- __ X__ _2 __ _4 _2 5 c _, - 1( b+C^X
Q9 -x1-(a+2bx+ cx( + c) dx,
hence by integration:
_ - 5 -- / x-n-2d 7 -- xn-1dx, dxrzlx
I. '1J- ------ + a2 n- J +nc-J R.
Putting for n the values 1, 2, 3... in succession, we have:
J: -v- = - dd x V_xd_ _c xc 3b {xdx aC d c
VR c c V2c 2cf JV/
fx:3dx X/J 5 _ 2fda f)uxdx
VR 3c - cJ VR cJ V3
( x2 5bx 15b-2 c- 4ac / 5 -- 3:-JC\ / X
3 c 6 C2 + 6c3 /1 2c / c y
In general:
11~, a dl _ Yv=n J8 I
IXnI. f dx.v dx
V 1