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.

~ 150.

IDefinition in case of infinite limits.

263

w w w w
W
— ' cos (v) cos (i2). 1 ]cos (x2) 7
_ COS ((t2) COS(62) 1 icos(X2) dz
2 wo 2 u 2 j -- *
The absolute difference of these two first terms does not exceed.
1
Further, since the function - does not change sign, we see by the
First Theorem of the Mean Value that the amount of:
w w
1 'cos (x2) M dx M l.7 1/
j — - X I- -d. x 2 J - - - 1 - is smaller than
2 J x2 2 Jx2 2 Lu W 2 tb
21 U
since 11 signifies a mean value of cos (x2), and is therefore a proper
fraction. Accordingly:
w
abs sin (xl) CX < 23-,
and tends to zero as u increases.
We have:
o00 C
fsin (x2) dx,= (d = -- - 1 / 2n; (see ~ 158.).
0
Another example is:
J -— b dx,
isin X
0
which can similarly be proved to be finite. It is still more instructive
to consider the following process: Taking k7 an integer and a < z,
let w == k -+ a be an arbitrarily great number, then:
w 2 t k7t kn+lt-a
J sin x. ( ~ + + sin x
0 0u ( k -)e i
The terms of this infinite series (for L = co) alternate in sign and
decrease in amount; for, comparing:
kit (k+i) 7
/ sin. I' sin x
-11- IX with J CIx,
(k -1) t k t
by substituting x = y + - and so making the limits of the second
integral the same as those of the first, we find:
s'in x in ( Xin 7
(k+l)+ t kn kt
j x - -d- ( -I Ynd
JI (k + (-J) 7 1Y l