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.

~ 138-140. Trigonometric and exponential functions. 237
Similarly the other equations are inapplicable in the cases of mn
or n being - 1. But here too the condition of being integrable
rationally holds.
The recurring formulas show that in all other cases the exponents
m and n can be brought down to numbers between - 1 and + 1, or
O and 2. If m and n are integers, we are in all cases led by repeated
application of the recurring formulas to one of the eight integrals:
fsin xdx = - cos x + C. Jcos xdx = sil x + C.
fi x Jd = (z) + C-= (tang x) + (~ 135.)
Eosx -- i y (tan (x + -) (y- + z).
J sinxdx cos xdx
J cos - -- (cos x) + C. - I(siunx) + C.
sin x cos x dx sin 2x dx= -4 cos2x + C.
dx
f- -- = — 2 dX =I(tangx) + C.
J sin x cos x 2 sin2 tangx)
140. Putting ekxdx d= d7 ) and integrating by parts we find:
1. fekxsinnxx = e lsin _ 7 e- k sinn-lxcos xc7x.
Likewise we find for this new integral:
kX sln-1 X x cos x d sin - x cosx (n —1)J ekxin-2 cos2 x dx
+ {fekxsinnxdx,
or, as cos2x - 1- sin2x:
2. ek-sin-1xcosxdx-0___ sin_ - x d x
e. jk sin-XcsX dX eksin cos (
+ - fekxsinn xd x.
If we combine equations 1. and 2. we have:
3. feksinxdx ekxsinn-1x(ksinx -ncosx)
+n(n -1) fk XSiln2 s d x.
+ 72 + n2
For n = — 1 this is:.sin d eX(ksinx - cosx)
J esnxdx == 7 k2 + 1