∫ √(1 − x2)dx
put x = sin z; the integral becomes
∫ cos z · cos z dz = ∫ ½ (1 + cos 2z)dz = ½ (z + ½ sin 2z) = ½ (z + 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 Integration in terms of elementary functions. 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 θ, x = a sec θ, x = a tan θ are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin θ and cos θ, and it can be reduced to a rational function of t by the substitution tan ½θ = 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)−1/2 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 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.)
∫ (x2 + a) − ½ dx = log {x + (x2 + a)1/2 }.
(ii.)
| ∫ | dx |
| (x − p) √(ax2 + 2bx + c) |
can be evaluated by the substitution x − p = 1/z, and