we deduce the equation

549

- iff- » lf' 1 3 (211-I) 'll

lx. ~" d = 52" J =;;; . .

f) 0 sm x x 0 cos x x 2 4 . 2n 2, (nan1nteger). f(x)dx = ae) gag. &

" "' - -» 5" . .

As an example' in the int?gral (XJ 0 Sm-»+1xdx= 0 COSzn+1xdx= »(n anrnteger). V (I x2)dx

Dllt 1'¢=SlH Z; the integral becomes 1 1 1 (xi.) (T§ @&?57, can be reduced by one of the substitutions fcos z.cos zdz=ji(I +cos 2z)dz= § (z-{-5 sin 2z) =;(z-{-sin zcos z). 49. The indefinite integrals of certain classes of functions can be by means of a finite number of operations of addition or multiplication in terms of the se-called “elementary " expressed

x:l';3;';" functions. The elerne ntary functions are rational algeterms of braic functions, Implicit algebraic functions, exponential ekmwh and logarithms, trigonometrical and inverse circular fum, functions. The following are among the classes of xt;-:Is functions whose integrals involve the elementary functions only: (L) all rational functions; (ii.) all irrational functions of the form f(x, y), where f denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv) all rational functions of ez; (v.) all rational integral functions of the variables x, e", e'>', sin mx, cos mx, sin nx, cos nx, . in which a, b, . . and m, n, . are any constants. The integration of a rational function is generally effected by resolving the function into partial fractions, the function being first expressed as the quotient of two rational integral functions. Corresponding to any simple root of the denominator there is a logarithmic term in the integral. If any of the roots of the denominator are repeated there are rational algebraic terms in the integral. The operation of resolving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic part of the integral can always be found without obtaining all the roots of the denominator. Reference may be made to C. Herrnite, Cours d'analyse, Paris, 1873. The integration of other functions, which can be integrated in terms of the elementary functions, can usually be effected by transforming the functions into rational functions, 'possibly after preliminary integrations by parts. In the case of rational functions of x and a radical of the form ~/(ax2~l-bx-l-c) the radical can be reduced by a linear substitution to one of the forms / (412-xi), /(262-a2), 1/(362-fl'-(12). The substitutions x=a sin 0, x=a sec 0, x=a tan 0 are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin 6 and cos 6, and it can be reduced to a rational function of t by the substitution tan § o=r There are many other substitutions by which such integrals can be determined. Sometimes we may have information as to the functional character of the integral without being able to determine it. For example, when the subject of integration is of the form (ax'-l-bx3~l-ox”-l-:lx-{-e)'l the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see FUNCTION). Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books of the integral calculus. In regard to the functional character of the inte ral reference may be made to G. H. Hardy's tract, The In§ tegralion of Funcnons af a Single Variable (Cambridge, 1905), and to the memoirs there quoted. A few results are added here (i) f(x”+a)-ldx =log {x-I-(xf-l-a)l}. (ii) (x PN <a(€;+2bx { C can be evaluated by the substitution x-p=I/z, and -<7J-iC-5-1— can be deduced by differ(x-p)

(ax +2bx+c)

entiating (1z- I) times with respect to p. (iii.) can be reduced by the substitution y' = (ax2-1-zbx-1-c)/(ax”+2;3x-l-7) to the form dy dy

A ~/(>*"3”)+B ~/ ot-m

where A and B are constants, and M and A2 are the two values of A for which (a->a)x2+2(b->B)x+c -M is a perfect square (see A. G. Greenhill, A Chapter in the Integral Calculus, London, 1888). (iv.) f.c"'(ax"+b) Pdx, in which m, n, p are rational, can be reduced, by putting ax“=bl, to depend upon ]l"(I-}1l)"dl. If p is an integer and q a fraction rls, we put t=uS. If q is an integer and p =1'/s we put I+t=u~*. If p-l-q is an integer and p=r/s we put I -l-l=tu“. These integrals, called “ binomial integrals, " were investigated by Newton (De quadrature cun/arum).

(v.) £5-£=log tang, fvi.) f£J%: =log (tan oc-l-sec x). (vii.) fe" sin (bx-l-a)dx = (az-[~b')"e“1{a sin'(bx-l-a) -bcos (bx+a)}. (viii) f sin'" x cos" x dx can be reduced by differentiating function of the form sinp x cos'1 x;

d sinx I gsin”x I—q q r

e'g' Z7 cos'1 x cos'1”1x +cos°+'x cos““1x+coS“+'x Hence

dx sinx +11-2 dx V

cos"x (n-I)cos"'1x n-I coS"'”x

e-l-cos x e-l-cos x

COS qi 1-l-ecos x' Cosh u I -1-ecos x of which the first or the second is to be employed according as e < or > I.

50. Among the integrals of transcendental functions Newtrggs. which lead to new transcendental functions we may notice gemieuts, x dx logag If

Y or U ci.,

0 log x xcalled

the “ logarithmic integral, " and denoted by “ Li x, ” also the integrals

X . 2-

sin x cos x

0—37 dx and no-?-dx,

called the “ sine integral ” and the “ cosine integral, ” and denoted by “ Si x ” and “ Ci x, ” also the integral x 2

-Z

edx

0

called the “error-function integral, ” and denoted by “Erf x." All these functions have been tabulated(see'l'ABx.Es, MArHEMATrcAL) 51. New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration integrals being fixed. Prominent among such functions are the ° Beta and Gamma functions expressed by the equations Eulerian

BU. ml = fy-1<1-~>m~1dx,

OO

I'(n) = e"l"“1dl.

0

When n is a positive integer 1'(n+I) =n ! The Beta function (or “ Eulerian integral of the first kind ”) is expressible in terms of Gamma functions (or “ Eulerian integrals of the second kind ”) by the formula

B(l, m) I'(l-l-m) =I'(l) I'(m).

The Gamma function satisfies the difference equation I'(x-l-1) =x1'(x),

and also the equation

I'(x) 1'(1 -x) =1r/ sin (x-/r),

with the particular result

Ui) = / 1r-The

number

~ illos Fil +x)} .or-1"(I).

dx ¢,0

is called “ Euler's constant." and is equal to the limit lim., .=o, , L (14-5-l-Q-l- ..-l-it -logn; its value to 15 decimal places is o-577 2r5 664 901 532. The function log I'(I+x) can be expanded in the series log r(r+x)=; log -s lOg'§ %;'l-{I'l'F'(I)}x "i(Ss"1>x3"ls(Sa-llxs*-~where

S2r+1= I "l';§ '.q'l';éq'l” - - ~,

and the series for log 1'(I -{-x) converges when x lies between - I and 1.

52. Definite integrals can sometimes be evaluated when the limits of integration are some particular numbers, although the corresponding indefinite integrals cannot be found. Definite For example, we have the result '”t"5""'1s° I

0 (I -x2)"% logxdx= *%7f'lOg2,

although the indefinite integral of (I -x”)"l log x cannot be found. Numbers of definite integrals are expressible in terms of the transcendental functions mentioned in § 50 or in terms of Gamma functions. For the calculation of dennite integrals we have the following methods:-

(i.) Differentiation with respect to a parameter. (i'i.) Integration with respect to a parameter. (iii.) Expansion in infinite series and integration term by term. (iv.) Contour integration.

The first three methods involve an interchange of the order of two limiting operations, and they are valid only when the functions

satisfy certain conditions of continuity, or, in case the limits of