00 
sin x? dx. [568 
) 
allies of cos x 2 and sin x 2 , 
•als of these functions. 
ueut to prove that the 
so, the double integrals 
nfinite square (or, if we 
le ratio having any value 
'ecedes, the values taken 
seen in a very general 
let the plane of xy be 
►, \/(37r), &c...., then in 
e maximum (positive or 
successive zones decrease, 
le successive zones; the 
series having no determ- 
'nto squares by the lines 
etween successive squares, 
iths continually diminish, 
f the succession of the 
inution in the values of 
alue of the integral for 
be, and which I assume 
this mode of integration 
ie square into indefinitely 
! interesting to carry out 
lat we actually obtain a 
then the quantity under 
m x 2 = yt to x 2 — 277-, and 
568] 
NOTE ON THE INTEGRALS COS X 2 dx AND sin X 2 dx. 
r • 
si] 
J 0 
59 
which, for r large, may be taken to be 
-if 
sin udu _ 1 
V(r7r) ’ d( r7r )’ 
viz. r being large, we have A r differing from the above value 
of the order . 
V(i’7T) 
by a quantity 
It is obviously immaterial whether we integrate from x 2 = 0 to (?i + l)7r or to 
(?i+l)7r+e, where e has any value less than 7r; for by so doing, we alter the value 
of the integral by a quantity less than A n+1 , and which consequently vanishes when n 
is indefinitely large. And similarly, it is immaterial whether we stop at an odd or 
an even value of n. 
We have therefore 
v = 
(»+1) ii- 
sin x 2 dx = A 0 — A 1 + A 2 ... 4- (—) n A n , 
or, taking n to be odd, this is 
= A 0 Ai + A 2 ... A n , 
or, say it is 
= (A 0 — Aj) + (A 2 — A 3 )... + (A n _ x — A n ), 
viz. n here denotes an indefinitely large odd integer. 
If instead of A 0 - A x + A 2 - A 3 + &c., the signs had been all positive, then the 
term A being ultimately as the series would have been divergent, and would 
have had no definite sum: but with the actual alternate signs, the series is convergent, 
and the sum has a determinate value. To show this more distinctly, observe that we 
have 
sin udy 
A A -I-V— 1 if' sin(ra ' + № ) rfM --if' 
A ’-' r ~ { > ’ 2 J-, -Ji-nr + u) ’ a J-. 
V(r7r + U) ’ 
or, taking the integral from — 7r to 0 and from 0 to 7r, and in the first integral 
writing — u in place of u, then 
A r _j — A r = \ I sin 
0 
udu -I 
(V (i’^T — U) \J{ r ^ + u )\ 
where, r being large, expanding the term in { } in ascending powers of u, then 
A r _ x - A r is of the order 4: and the series (A 0 — AJ +(A 2 — A 3 ) ... + (A n _! — A n ) is 
therefore convergent, and the sum as n is increased approaches a definite limit. Hence 
the integral v has a definite value: and similarly, the integral u has a definite value. 
8—2