Page:EB1911 - Volume 14.djvu/580

OUTLINES] we deduce the equation

$\int f(x)dx=\int \phi(z) \frac{dx}{dz}dz.$

As an example, in the integral

$\int \sqrt{}(1-x^2)dx$

put x=sin z; the integral becomes

$\int \cos z \cos z dx=\int\tfrac{1}{2}(1+\cos{2z}=\tfrac{1}{2}\left(z+\tfrac{1}{2}\sin{2z}\right)=\tfrac{1}{2}(\sin z \cos z).$

49. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called “elementary” functions. The elementary functions are rational algebraic functions, implicit algebraic functions, exponentials

and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form ƒ(x, y), where ƒ 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 ex; (v.) all rational integral functions of the variables x, eax, ebx, 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. Hermite, 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+bx+c) the radical can be reduced by a linear substitution to one of the forms √(a2−x2), √(x2−a2), √(x2+a2). The substitutions x = a sin, x = a sec , x = a tan are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin and cos, and it can be reduced to a rational function of t by the substitution tan = t. 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 (ax4 + bx3+cx2+dx+e)− the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see ).

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 integral reference may be made to G. H. Hardy’s tract, The Integration of Functions of a Single Variable (Cambridge, 1905), and to the memoirs there quoted. A few results are added here

(i.) $$\int(x^2+a)-\tfrac{1}{2}dx = \log \left\{x+ (x^2+a)^{\tfrac{1}{2}}\right\}.$$

(ii.)$$\int \frac{dx}{x-p)^n \sqrt{}(ax^2+2bx+c)}$$ can be evaluated by the substitution x-p=1/z, and $$\int \frac{dx}{x-p)^n \sqrt{}(ax^2+2bx+c)}$$ can be deduced by differentiating (n−1) times with represt to p.

(iii.)$$\int \frac{(\text{H}x+\text{K})dx}{(\alpha x^2+2 \beta x+ \gamma) \sqrt{}(ax^2+2bx+c)}$$ can be reduced by the substitution $$y^2=(ax^2+2bx+c)/(\alpha x^2+2 \beta x+ \gamma) \,$$ to the form

$\text{A}\int \frac{dy}{\sqrt{}(\lambda_1-y^2)}+\text{B}\int \frac{dy}{\sqrt{}(y^2-\lambda_2)}$

where A and B are constants, and 1 and 2 are the two values of for which (a−)x2+2(b−)x+c− is a perfect square (see A. G. Greenhill, A Chapter in the Integral Calculus, London, 1888).

(iv.) ∫xm (axn+b)p dx, in which m, n, p are rational, can be reduced, by putting axn = bt, to depend upon ∫tq (1+t)pdt. If p is an integer and q a fraction r/s, we put t = us. If q is an integer and p = r/s we put 1+t = us. If p+q is an integer and p = r/s we put 1+t = tus. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).

(v.)&emsp;$$\int \frac{dx}{\sin x}=\log \tan\frac{x}{2},$$ &emsp;&emsp;(vi.)$$\int \frac{dx}{\cos x}=\log (\tan x+\sec x).$$

(vii.)&ensp;∫$$e^{ax} \sin (bx+\alpha)dx = (a^2+b^2)^{-1}e^{ax}\{a \sin (bx+\alpha)-b\cos(bx+\alpha)\}.$$

(viii.) ∫$$\sin^m x \cos^n x dx$$ can be reduced by differentiating a function of the form sinp x cosp x;

e.g. $\frac{d}{dx}\frac{\sin x}{\cos^q x}=\frac{1}{\cos^{q-1}x}+\frac{q \sin^2 x}{\cos^{q+1}x}=\frac{1-q}{\cos^{q-1}x}+\frac{q}{\cos^{q+1}x}.$

Hence

$\int \frac{dx}{cos^q x}=\frac{\sin x}{(n-1) \cos^{n-1} x}+\frac{n-2}{n-1} \int \frac{dx}{ \cos^{n-2} x}.$

(xi.)$$\int_0^{\frac{1}{2}\pi}\sin^{2n}xdx=\int_0^{\frac{1}{2}\pi}\cos^{2n}xdx=\frac{1 \cdot 3 \ldots(2n-1)}{2 \cdot 4 \ldots 2n}\cdot\frac{\pi}{2},$$  (n an integer)

(x.)$$\int_0^{\frac{1}{2}\pi}\sin^{2n+1}xdx=\int_0^{\frac{1}{2}\pi}\cos^{2n+1}xdx=\frac{2 \cdot 4 \ldots2n}{3 \cdot 5 \ldots (2n+1)},$$  (n an integer)

(xi.)$$\int\frac{dx}{(1+e \cos x)^n}$$ can be reduced by one of the substitutions

$\cos \phi=\frac{e+\cos x}{1+e \cos x}, \quad \cosh u=\frac{e+\cos x}{1+e \cos x},$

of which the first or the second is to be employed according as e 1.

50. Among the integrals of transcendental functions which lead to new transcendental functions we may notice

$\int_0^x \frac{dx}{\log x},$ or$\int_{-x}^{\log x}\frac{e^z}{z}dz,$

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

$\int_0^x \frac{\sin x}{x}dx,$ and$\int_{\infty}^{x}\frac{\cos x}{x}dx,$

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

$\int_0^x e^{-x^2}dx,$

called the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (see ).

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 being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equations

$\Beta(l,m)=\int_0^1 x^{l-1}(1-x)^{m-1}dx,$

$\Gamma(n)=\int_0^{\infty} e^{-t}l^(n-1)dt.$

When n is a positive integer (n + 1) = 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

$\Beta(l, m) \cdot \Gamma(l+m)=\Gamma(l)\cdot \Gamma(m).$

The Gamma function satisfies the difference equation

$\Gamma(x+1)=x\Gamma(x)\,,$

and also the equation

$\Gamma(x) \cdot \Gamma(1-x)= \pi / \sin{x \pi},$

with the particular result

$\Gamma \left( \frac{1}{2} \right)=\sqrt \pi.$

The number

$-\left \lbrack \frac{d}{dx}\left \{\log {\Gamma (1+x)}\right \}\right \rbrack_{x=0},$ or$\Gamma^\prime(1),$

is called “Euler’s constant,” and is equal to the limit

$\lim_{n = \infty}\left \lbrack 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}-\log n \right \rbrack;$

its value to 15 decimal places is 0.577 215 664 901 532.

The function log (1 + x) can be expanded in the series

$\log \Gamma(1+x)=\frac{1}{2}\log \left(\frac{x\pi}{\sin x\pi}\right)-\frac{1}{2} \log \frac{1+x}{1-x}+\left \{1+\Gamma^\prime(1)\right \}x$

$-\frac{1}{3}(S_3-1)x^3-\frac{1}{5}(S_5-1)x^5-\ldots,$

where

$S_{2r+1}=1+\frac{1}{2^{r+1}}+\frac{1}{3^{r+1}}+\ldots,$|undefined

and the series for log (1 + x) converges when x lies between −1 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. For example, we have the result

$\int_0^1(1-x^2)^{-\frac{1}{2}}\log xdx=-\tfrac{1}{2}\pi \log 2,$|undefined

although the indefinite integral of (1−x2)− 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 definite integrals we have the following methods:—

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