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.

~ 117 ---119.

The binomial integral.

199

Vb -a P m- - " X (laz (bxfl)iP+1
'id-V.. + bx..)~ dx =- an (p + )
a 4- n(p_+ )
vi. -1(a + bXn)p + dIX ixm(a + x )P ( b
J ax 1 ( +Ixam
( am b j " +n —1 (a + bxfl)P dx.
These formulas of reduction become useless when the quantities
in the denominators vanish; but in all such cases the integral becomes
rational by the substitutions already assigned; the only cases worthy
of special notice here are n = 0 and np +- m == 0; for these:
f -(a - - clx 1- _ -a dl, (a + bx );
(IX dz (at + b x;
x Inj -a
expressed by algebraic functions and by an integral in which m the
exponent of x is between zero and n, and therefore the ratio nm: n
between 0 and 1.
The forms III. and V. show that the binomial integral can be
expressed by algebraic functions and by an integral in which the
exponentp is likewise a positive fraction between zero and 1.
Finexp d by all gebr aic functions and by an interacl in which
Finally if neither vanish nor the two fractions and p
supplement to unity, the value of the integral can be expressed only
by an infinite series, by expanding in a series of powers the binomial
of the function to be integrated, and then forming the integral of this
series multiplied by xm-l (Chap. IV).
119. The group of irrational functions next in order consists of
those in which the square root of a polynomial of the second degree
enters *):
jF(x, 1/R)dx, i = a + 2bx + cx2,
where F means a rational combination of x and J/R. The sign of
j/E is to be taken positively unless the opposite sign is prefixed.
Arranged by its rational and irrational parts we have:
*) Euler: loc. cit., Cap. II, ~ 88-93.