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.

286 Examples on the calculation of definite integrals. Bk. III. ch VII.
This investigation was necessary, because the integrals
coo Go
dy, Y
0 y, ^y(1+y)
are not separately finite.
Accordingly we have:
VI) dlog F(o) _ -_ - d ) dy.) e
da
dec V(1J- +y)
Differentiating with respect to a and then putting 1 + y =- ex we find:
d' 2I F a) 1(1+ y) X- ax
VII) da =j I, dy = - dx (a > 0).
This is an integral adapted to integration by expanding. We
have:
I - (n+l)x
- 1 + e- + e-2x + e 3x +.. e-nz + 1, n - - e-z 7
therefore:
0
j Lz e ax ax
= f xea dx +J exia+)dx + (... 4f+J xe-(+I^dx +J nX e d x.
0 0 0 0
vo o
Now:
(a + n) -(a ~ n)2 (a + n)2
0 0
and we can choose n (cf. also ~ 130), so that
/~p(a+n+l)x
co,
shall become arbitrarily small, because:
1 1 1
fx~-(a —n-1l)x dx =M Mf-(a+n+l)x dx 2= M)
l _ e-x a ( n -- +1
oo o CO
re-(a+nd-1+l)a a -(a+n+l)x -(a+n+l)\x
cl --- — x M X e -- (a+n+l)xd I -X x -_-c-' Xe
J 1-e- J \ xa+t+ I (a+4-t+-l)-2
where MI denotes a mean value of in the interval from 0 to 1,
1 - e-x