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.

~ 124. Three fundamental elliptic integrals. 211
so that the reduction to the first two fundamental integrals is
effected.
Putting the values 1, 0, -1, - 2,... for n in the same recurring formula, it enables us to express integrals of the form -xnVx
by means of the first two fundamental integrals and by the integral
J'x. X To show this, let us put - n + 3 for n; we find:. ^-1 - a(2 - i)J- d - b(n - 4)J 4
YW d x dx
- c(n - 2)J J - d(n - )j
Now provided a does not vanish, this gives for n = 2, 3, etc.:
cdx _ -p b r /Z rdx
J xsV2 ~~ a x 'c^J xz K 2aJ vt?
dx 1 VR 3 b dx c dx dd CdXL
XJ VP 3 ~2a 3 J xy4Ra - J YR fx AJ I V1
In general for n > 2:
v=n —1
dx / a, i dx dx + + dx
dx ad
In case a vanishes, XJ v is expressed by the first two fundamental
integrals, as appears from the recurring formula II..
Formula I. admits of generalisation according to the method already
applied (~ 121), by writing for x, x- Q and for a + bx + cx2 + dx3,
A + Bx + Cx2 + Dx3, where:
a =A + OB + 2C +- /= f-( 9),
b B + 2C + 32D f'(9),
c = C + 3 D -= Jjf"(),
d = f"' ().
Thlen if we imagine the letters a, b, c, d, put for A, B, C, D),
we have:
11r. (x V — f(e)(i,- 1 d x _ /'(9) (_,,1 _ e_/
dx) f i dx
dX, Xy ~+/ - 1/ dx
(x- f() ( - 2)J (x - x - V.
f_ '(x -- Q)dx
14'