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.

~ 154-155. Applications of the methods. 275
2a) -x a c a
3 a) e-X-e dx = d (a).
155. Second group. (~ 140.)
00
Ce-ax sin bxcdx
Je-axcos b x d x — = a2 - b
-ax COS b X d —
0
Hence follows by differentiation with respect to a:
faxXS ab a2 - [)2
0o o
2) je- x sin b xcidx= 2 '
Again by integration with respect to a between the limits a and,B:
e-a"x- _e — x a
faxe e — sin b xdx = tan-' - tan-'
of x b b
3) (a and /3 > 0).
fe-aX- e b d 1 2 b2
e -- cos bxdx -- 1 —,
J x 2 a2 + b
Making a converge to zero in the first of these two integrals, we
finl:
0
J4)- e sill bxdx ~ tan- b '
For, the integral 3) is a continuous function of a inclusive of the
value a == 0. In fact, separating the integral from 0 to w and from
tv to oo, as ~ 152 requires, we can evidently by choice of w alone,
independently of the value h, make:
00
I f -- sin bxdx
w
less than an arbitrarily small number, because (~ 150):
GO
Jsinb x
w
18'