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.

280 Examples on the calculation of definite 'integrals. Bk. III. ch. VII.
therefore:
o0 O
(6) tSin da = 7/. We likewise find: jCV- c =... )
159. Sixth group.
We proved in ~ 115 the formula:

1)- e l- ai
/X —lo'dx 7_ t _ m e n
l) 7 an j- i =.. -
y x iD.sin -- 7r
v^./ x"-t- e~~~~~~~~~~~l

(m and n positive integers, m < n)
- -r < c < + r.

Putting xn = z and denoting the rational fraction "' by a, we find:
sin
2),0. e(a-ai (O < a < 1,2 - < a < +

0e

Now this equation was proved only for rational proper fractions.
But since the definite integral, as well as the function on the right
side, is a continuous function of a (proof as in ~ 152), the equation.
is still true for every irrational number less than 1. For a =-0
we find:

3)

o x + 1 sin a 7f

(0 < a < 1),

or, separating the integrals between 0 and 1, and between 1 and oo,
and introducing into the second - instead of x:
zn - ~~x

1
xa -ldx+
J x-+7Y

r-. a _I_ _a-x
J + -J I+ Al.
J o
d ex d

rx-ad0
0

4)

— 1 + X- a
x + 1 — sin am

(O < a < 1).

Although integral 1) ceases to be finite for a = + z, because the
function to be integrated becomes infinite in the first order at the
point x= 1, still the integral
r in -- 1- -
J n+ C-e C
must exist even for a = z, because the factor x - 1 cancels in
*) The problem, of determining curves having the radius of curvature in
inverse proportion to the length of the arc, conducted Euler to these integrals.
(Salomon's Uebersetzung der Euler'schen Integralrechnung. Vol. IV. Suppl., p. 321.)