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.

~ 131-133.

Series for an arc in terms of its sine.

229

finite value I~ although the function to be integrated becomes infinite at the upper limit.
133. The elliptic integral:
j1/(i X)(1 -.lc2x2) d — /-sin2)
o o
can be developed by powers of k. Expanding:
~(1-k' sin) _ 1.3 1+- sins'np_- 1.3.5
(1-7X-s1ng) = 1 &2 - sin2 gn +2 2^s- 3 g+ 2.4 6 3G
we have therefore:
1,* (p qpa
F(p) =sj + sin gd + 1 3 7tsind
JI -k2ifsn'pdcp 24 js 0II4p
% 0 0
1+ '3.5 716 Jsinp +.. (k2 sin? q < 1).
The integrals in this series belong to the binomial integrals
investigated in ~ 118 as is seen by substituting sin p - x, they are
determined by integrating by parts; we have:
sin2M pC = -J sin2- q p d(cos p)
-- - sin2~-1'Ipcos p -+ (2m - 1) sin2-2gpcos2lpdp,
hence replacing in the last term cos2qp by 1 -- sinl2, transposing and
inserting the limits we find:
0 O
For the limits zero and I r we have:., 2m-1 * - 7 2 - 1 2 m-3 2 m-5 3 1 t
fin n 2 ==p I = -7 " Si2m -2(pd ==T -- --
J sinsm'pclp= 2 s in f 1'pd 2 m 2n m-2 2mn-4 4 2 2
0 0
Hence the "complete integral" is:
V
2 A r)- apo- 2 1 + 2 2 D.
0
Similarly we obtain for the integral:
(p
E(') fALpd(p = (p) - 2Z('p)
by expanding:
2 1 I 7 2 -2.14Sln —2.1.36