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.

192

The integral of rational algebraic functions. Bk. III. ch. II.

But,
fTwi- di1 d if >1, and
J x - z- 1 (X-)~ - 1
hence:
j-2,1 - 2
III. I )Idx 1-I

' dX (x,
/x -- I (x- a),
- - aO. - A-i-_,l(x - c)

J p(x) A~ -- — j 1 (x a)- - 1 - I..
j=O
+........
h e ' -j-l any rtinl a ic ntin x c b
The integral of cany rational algebraic fiunctio cn ca be expressed by a combination of rational algebraic functions and logarithms;
to integrate it explicitly the vanishing values of p (x) must be determined;
moreover circular functions can be introduced instead of the logarithms.
Example:

If we put x =
p(- 1+h)
9)q (- + +h)

V (X) 2x4 - 3x- + — 21
(x) (x + 1)()3 - 2)2
= -- 1 + h, then
27 -18 h+-21h2 - 11h 7+- 2 h4 h 2 7
h -- 3 + 2h +- 9_6 h + _2 2
9 _ 6h,_h9 - 6h

p (X) 3 2 1
cpx) ( + I)3 ( 1) (X + 1) ( - 2)2
d- 1 1
gq(X) ( o (x i- 1) + + 1) + '
114. It is also possible in the general case just treated to prevent complex quantities appearing in the ultimate result, if, when
(p has complex roots, the coefficients of (p and 7p are real; here
again the partial fractions belonging to conjugate complex roots are
to be combined. We start directly from the theorem:
When p (x) = (x- a=2 -- P2).,i(x), there is only a single way
in which to effect the resolution:
ip(x) PoX + Qo __ (x)
P () (X -_a2 + P2) ^ (xca2 + 2)-1 -I) (x) 7
where Po and Q0 signify real constants.
We must have

i. e.

V(x) -= (P0x + Qo),) (x) + 1 (x) (x -- U + P3),
Po(c + i ) + QO = (+ t iB *

Accordingly:
(X)__ P(X +Oo P x + Q,..+P xQ + + _ (X)
(W (3)+ ( _ _2 _ P2)_ (X 2 _+2)R-1 _ 9(P1 ) _