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.

~ 136-138. The Logarithmic Integral. 235
Forming the function:
p(x) (e-a- e- ),
-X -x
its derived is p(x) =- Now since F'(x) =, e (x) is> F(x)
if x > a. Hence the functions F(x) - F(a) and p(x) both increase
continuously from zero for x > a, so that the second function is
always greater than the first; hence
(~)- > F(o) - F(a),
therefore
F(oo) < F(a) + -
Thus the error incurred in putting C =( - F(a) is less than C-a )
In this manner, assuming ex. gr. a = 10, we determine the value of
C from that of F(10) with a defect less than
-10
e < 0,00001.
The value of C, the Eulerian constant, for which we shall give
a more rapidly convergent series ~ 165, is: 0.5772156649...
138. Integrating by parts:
Jxmcosxdx= x'cos (x - 7) - m f xmlcos (x - -r) dx,
jf xsil xdx = il ( -- n Jx,1 sin (x n -(x ) dx.
Hence for a positive integer m:
X~ 2-sin1) xrn-lm V 1 I
v=o
v ---0
fsinxdx cosi 1 n sin xdx
Xt (_ - 1)_'. X
/sinxdr sinl 1 pCosxdx
XeJ (m - 1) Xm-l1 -J
In this case a positive integer value of n leads to the integrals:
/ cosxdx f sinx dx
that can only be found by expansion. We get:
') Minding: Handbuch der Differential- und Integralrechnung, p. 191,