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.

Fifth Chapter.
Integrals of transcendental functions.*)
135. If f(ex) denote a rational function of ex, ff(ex)dx is
dz
transformed into a rational integral by substituting z = c, dz i dx.
The integral of a rational function of sin x and cos x can be
converted into one of the above form by substituting:
eix- e —ix ix c-ix
4+ e-^. e- 6
cos x, sin x - 2
and therefore also into the integral of a rational function.
However as this introduces imaginary quantities, we ordinarily
prefer to substitute:
* 21 i - z 2dz
tanllx= —, sinx ] +z cosx=l dx-,-dx
t 1 + Z2 2 = Z2 7+ Z2
Since Jxdf - xf-f- fdx we can- also calculate Jxdf by the
rule of rational functions, when f is any rational function of sin x
and cos x.
136. By partial integration:
Jexxm d x ==- xmex -nJxm-lexdx
f dx -- - 1 i r dx
J;;X z-1 ] i _ 1M dx.
If m be a positive integer, we obtain by the first formula:
jexxm dx - Im e -- (- 1)v X'-
ito
The second formula when in is an integer leads to the equation:
Jd tv (m- -__ dx.
__dx 1x ( — + e- dx.
v=1 LV(m1)yXrnv X x
*) Without entering on a general investigation, under what conditions the
integrals of transcendental functions can be evaluated in finite terms, we only
collect in this chapter those formulas to which the simplest applications of
analysis lead us. Euler: ibid., Ch. IV and V. General investigations of these
integrals were' given by Hermite: Cours d'Analyse, p. 320.