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.

214

The integral of explicitly irrational functions. Bk. III. ch. Ill.

JF() dx = PJ (P 1+A);zf z -=(AS2 + c) (AI2 + C'),
where:
A =(q - a) ), C), A'= (y - y) (q -- O'),
C'==(P-Y)( P-6' A(a - 3- y a) a-t(r ) + - y( a (- +P)
C'=(p- )(p- 6),P+ -Jq= + )_(7+, p = 2 ((- +) - (v + ))
All these quantities are real, when the coefficients in P are real.
Developing the rational function F (P- +qz) we can collect the
odd and the even terms in its numerator as well as in its denominator:
F G(z) + zH(z)
Gj (Z2) + z Hi(z2)
If we multiply this above and below by G1 - 2H, we get in the
denominator only even powers, so that:
~F mq (z2) Z V (z2)
-- +
2_ _ LI,S^
Vz Z Yz
The integral of the second term is converted by the substitution z2 - t
into an integral of the form:
1 r fv dt
AJ v +(At+ c A't+C')
and is therefore evaluated by logarithmic and algebraic functions.
In the integral of the first term the polynomial Z can be brought
to the form:
(1 - y2) (1- - y2),
where 0 < k2 < 1. In order to examine this, let us write:
- C' (1 + Z 2) (1 + A') L + 2) (1 + 22),
taking a2 > j2. Of course a, /, y here are not to be confounded with
the former notation for the roots of P. Now according to the signs
eight cases arise; in each we consider those values of 2, with values
corresponding for y, for which Z remains positive, so that both the
function to be integrated and the integral function are real for real
values of the variable.
1) Z-= -+ 7(1 + a222) (1 + P 22) remains positive for all real
values of a. Let us put:
ocz Y dy
Y _ -2 - (1 - y)
If 2 increase from - oo to 0, y increases from -- 1 to 0; and if z
increase from 0 to + oo, y increases from 0 to + 1; thus we have:
dz dy k ____2 a-2 ~2
V2 (Y Y 'V 0- -_ - 7
vZ ay rtom - )1 7 y) ) a2
The radicals in this equation as well as in all the analogous results
from 2) to 8) have the same sign on both sides.