122-124]
DERIVATIVES AND INTEGRALS
233
122. Transcendental Functions. Owing to the immense
variety of the different classes of transcendental functions, the
theory of their integration is a good deal less systematic than
that of the integration of rational or algebraical functions. We
shall consider in order a few classes of transcendental functions
whose integrals can always be found.
123. Polynomials in cosines and sines of multiples of x.
We can always integrate any function which is the sum of a
finite number of terms such as
A (cos ax) m (sin ax) m '{cos bx) n {sin bx) 11 '...
where m, m', n, ri, ... are positive integers and a, b, ... any real
numbers whatever. For such a term can be expressed as the
sum of a finite number of terms of the types
a cos [{pa + qb + ...) x], /3 sin {(p'a + q'b + ...) x]
and the integrals of these terms may be at once written down.
Examples LIII. 1. Integrate sin 3 # cos 2 23?. In this case we use the
formulae
sin 3 x=J (3 sin x - sin 3#), cos 2 2x=^ (1 + cos Ax).
Multiplying these two expressions and replacing sin x cos 4x, for example,
by J (sin 5x — sin 3#), we obtain
J(7 sin x — 5 sin 3x + 3 sin 5x — sin lx) dx
= - T 7 (i cos x + -¿’-g cos 3# - Aicos 5#+x fa cos *~ iX -
The integral may of course be obtained in a different form by different
methods. For example
Jsin 3 xcos 2 2xdx= I (4 cos 4 # - 4 cos 2 #+!)(! — cos 2 #) sin#d#,
which reduces, on making the substitution cosx—i, to
/
(4i 6 - 8t 4 + 5i 2 — 1) di—f cos 7 x - f cos 5 #+f cos 3 # — cos #.
It may of course be verified that this expression and the integral already
obtained differ only by a constant.
2. Integrate by any method cos ax cos bx, sin ax sin hx, cos ax sin hx,
cos 2 #, sin 3 #, cos 4 #, cosxcos2#cos 3#, cos i# cos 2#, cos 3 2#sin 2 3#, cos 5 #sin 7 #.
[In cases of this kind it is also sometimes convenient to use a formula of
reduction (Misc. Ex. 40).]
124. The integrals Jx n cos xdx, j’x n sinxdx and associated
integrals. The method of integration by parts enables us to
