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.

Seventh Chapter.
Examples on the calculation of definite integrals. The fundamental
formulas of Eulerian integrals.
154. First group.
t i1
1~) J.~a —1d () - -, ( — >0)
V 0
By successive differentiations with respect to the parameter a we obtain
the following integrals:
I
2) Jxa-1 x ()dx =-,
0
1 1
a- (lx) dx 1.2 x.. (jx)_- _ 1 )n- (-
o 0
By the Theorems ~ 152b and ~ 148, all these integrals hold for a > 0.
By integration with respect to the parameter a between the limits
a = a and j3, it is found that:
f 1 1 i F i
fda x -dx = fdx fxlda = x j - I ().
a 0 0 a 0
This formula holds good, provided a and /3 > 0.
Xa- 1
For, so long as a is positive, we have not only xa~- but also -
1(x)
integrable at the points x = 0 and x = 1 moreover the condition
stated at the end of ~ 153 is fulfilled.
Putting a = 1, we find:
1
3) J dx l(X7 1(), ( > 0).
0
Substituting in these integrals the function e-~ for x, the limits
become oo and 0, instead of 0 and 1, and we obtain for a > 0:
l a) -cdx ---
0