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.

198

The integral of explicitly-irrational functions.

Bk. III. ch. III.

When is an integer, the first expression becomes rational by
substituting S =- ti q being the denominator of the fraction p.
When p 4- "- is a n integer, the second expression becomes rational
by substituting:
1 -- az
b 1
118. Now although it is only under these determinate conditions
that the binomial integral can be expressed by explicitly irrational
algebraic functions and by logarithrms'), yet in all cases its integration
can be reduced to that of certain simple forms.
Consider the differential x~L-1(a + bx")PCdx as a product, either of:
-) (a + bx~') of, or of: ~+. 1- X-'
m b b J(np -- 1) ' -,
then integrating by parts:
'I. J -xl(a + bx')dx=. -...(...b -- n)- - (:a + b x)P-'ldx,
- "(a[bxn)P+' M __ It
II jX ( bxT b (p+ 1) it("b(Ri l)Jl r ----l (a+- bxn)'P1-lx.
We can alter these formulas so that in each integral on the right, one
exponent only shall be changed. In equation I. let us write:
Jnt+n-1 (a + bx)1 - I-(t +X jJI-(a j -bxn)P dx - ax — (c6-a+bxr)P-1dx,
also in equation II.:
J X-ln- l(a+- bxn)P+1 (i=af x;- n-l (a. bxn P dxn d — xm_ —l(a+bxn)Pc(x,
and combine in each case the integral on the right side with the
equal integral on the left; we find:
IV. j 4 (a - ~ bxdxndx- (M b a + Jbx '-n)P+- x
IV. "x-(a + bxpdx -- ( +bxn)P
d (fp)-F t - + In6 b
(in -n '? a -
-(ip +)J- m) b (I -+ x1))P (cx.
Thus the formulas III. and IV. diminish the exponents p and m
by the quantities 1 and n. These formulas can also be regarded as
reducing the integrals on the right to those on the left. For uniformity let us solve each equation for the integral on the right, having
first put p for p - 1 in III1 and m for m - n in IV- the results are:
*) Tschebychef: Sur l'int6gration de diff6rentielles irrationnelles. Liouville
Journal, T. XVIII. 1853.