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.

276 Examples on the calculation of definite integrals. Bk. III. ch. VII.
can be arbitrarily diminished by choice of w, and moreover by the
Second Theorem of the Mean Value we have:
fe-hxsin bx xin bx
0C ---- - dx = l' I -- dx,
w. w
the upper limit uz denoting a number between w and co. In like
manner, integral 4) is a continuous function of Tf. Therefore making
p become infinite, we obtain:
CO
5).f sinl bxcx -sn
5) (I dx= +
x 2
according as b > 0 or < 0. But this definite integral, as already
remarked, is not a continuous function of b when b = 0; for, at that
point its value is zero.
The formula 5) can be expressed generally. Replacing b in it
successively by b + a and by b - a and assuming that b > a > 0,
we have:
00 o
Jsin (b +- a)i -c f sin (b- a)x (x 2
xJ ~" J x'X 2.
0 - 0
Hence also, as we find by addition and subtraction:
cn cJ
COx 05~ --- —-------- dx = O (b r
silnbx cos ax d rsinax cos bx
XJ - dx-., J — z dx 0 (b a),
0 0
or:
00
2 sinbxcosax
6) ------ dx = 1 or =-0, according as b > or < a.
nj x
0
CO
1 sin 2bx
For. a = b:- -- dx= 2
x 2
0
156. Third group. (Laplace's integrals.)
0O
rinT bx cosax
Writing ----- ( and b > 0), we have - =0
0
when b < a, and t = -= z when b > a. Multiplying both sides by
e- b, where c is positive, and integrating from b = 0 to b == oo,
we find:
C/O COi e ' sin bx cos ax d
J tebcdb J e-bcdb - cb x.
o0 0
Interchanging the order of integrations in this integral, it becomes: