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.

~ 155-157. Laplace's integrals. 277
odx a os ax dx
o0 0
Hence:
0
f w ithsp --- t, te rut s:C
ss ax tru ' *)os saf x, 7 + (according as a > 0 or <0)
Cx2 J _ cX this is true also for a = 0.
If we differentiate with respect to a, the result is:
0
28 z sin nxdx _ + 7f C (according as a >.0 or < 0)
This differentiation with respect to a is permitted (~ 152); but a
further differentiation under the integral sign with respect to a is not
possible, because the derived function is no longer integrable between
the limits 0 and co. Integrating equation 1.) with respect to a between
the limits 0 and a positive, we have:
a),, a
f illX x2 (- a = 2c' ( -aC), (a > ~)
0 0
on the other hand for a negative:
o a
sina I eCZ\ n
J xc2 + ) x = 2 = 2 (cac _- ) (a < 0)
0 0
Successive differentiations with respect to the parameter c can also be
effected on integral 1).
157. Fourth group. (~ 139.)
silt l = - 1 XSill"2 +n dX
Iu~xdx === - -— snn -- 1 XOX --- n xud XxdxX
sin'~x d x.- -7 — I sin- s x,.
0 J
When m is an even integer = 2m, we have as il ~ 133:
in2 9n, 7 2X ( - 1 2w- 3 3 1 r
/ '2 ~~ 2m 2 m -;2 4 2 2
0