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.

236

Integrals of transcendental functions.

Bk. III. ch. V.

X7 2 X4 X 62
JCosXd I(X)- 212 + X- 61 + + ( 1)2n* 2,
sin X3 X5 37 X2n+1
rin etc.3
dX x 3 + 5 -717 - + + (- 1) -112n +
The first holds for every interval that does not include the number
zero, the second without restriction. Moreover, determinate values
must result even for infinite limits (~ 155).
139. The integralfsinmxcosnxdx is converted by the substitution:
sinx - - z~ ~ cl z
sinx==^, cosx=(1 —z, dx= -z- -
Z (1 - z)E
into the binomial integral:
r m-1 n-1
J2- (1 -g) 2ds,
The Theorems in 117 show that this integral can be brought
to a rational form when one of the numbers:
~ (m -1) - 1), -1 (- + n)
is integer; one of these equations must be fulfilled if mi and n are
both integers. In other cases the recurring formulas of ~ 118 are
applicable to the present integral. We can write down these six
recurring formulas retaining the trigonometric shape, directly thus:
I. fsi.no+ xcosn-x xn — fsin+2 X cosn- 2 xx
II. Jsnmxcosnxdx== Silm lxcos; +1 X sin-2 cos +2X.
J. nc-+- 1 n. + 1
II. Jsinmxcosnxdx= = sin n- i cOs + — lfsinm-_xcos+~2xclx,
Putting on the right in the first equation sin,,+2x = sinmx(l - cos2x),
and in the second cos+2X =- cosnX(1 - sin2x), we find:
III. jsinmx Xcosx im+l s +sin mxosn -1+ xdx,-
IV.Jsinm x cosn x dx = - +n xn -2XC`;
*l. wnm + is m+n n J
F. m X OSn X (_ Sinm' X Cosn+iXlit - i - 2
IV. sinm xcosxdx inm-=sn + + s nxcos dx.
If we solve these equations for the integrals on the right, and replace
in III. n - 2 by n, and in IV. m - 2 by n, we obtain:
J sinnm+l + xcs+x
VI. JsinmnxcoSnlxdx-=- sin + + in +2xcosnxdxx.
VI.j'sina x cos- x dx = sin"m l + x cosn ^ _ sin2- cos" x dx.
mn +1 + n+ I
Equations III. and IV. cannot be employed when n + n -- O. Here
we have for n - - m == --
s ( ingx tan = vd
oSX dx (tan x)v dx = v + 2, putting tan x =.
\co -b-/ dx