CHAPTER II. CONSTRUCTION OF RULES FOR INTEGRATION OF ALGEBRAICAL FUNCTIONS. 10.] We propose in the present and next Chapters to construct WE the rules of the Integral by inverting those of the Differential Calculus and first we shall from this point of view exhibit the forms of indefinite integrals which correspond to those theorems on definite integrals which have been proved in Article 8. THEOREM I. Since d'aF(x) = ar'(x)dx; .'. fax (x)dx = ar(x), = = a/F'(x) dx; (1) that is, in the integration of an infinitesimal element one of whose factors is a constant, the constant may be placed outside the sign of integration. ... THEOREM II.-Since d. {F(x) ±ƒ(x) ± ... } = d. F(x) d. f(x) ± ... = F(x) dx + f'(x) dx ± ...; ± f{x (x)dx ±ƒ'(x)dx ± ... } = F(x) ±ƒ(x) ± ... (2) that is, the integral of an algebraic sum of infinitesimal elements is equal to the sum of the integrals of the several infinitesimal elements. [{F'(x) +√/=1f'(x)}dx = F(x)+√/=1 f (x). THEOREM III.-Since · d.F(x) × ƒ (x) = ƒ (x) × F′(x) dx + F(x) × f'(x)dx; f(x) × F(x) dx = d.(F(x) × f(x)) − F(x) × ƒ'(x)dx ; ·. [f(x) ×x′(x)dx = x(x) × ƒ (x) – [ v(x) × f'(x) dx ; (3) which is the theorem in the Integral Calculus corresponding to that of the differentiation of compound functions as explained in Art. 31 of Vol. I, and which may also thus be proved: Let f(x) = 2; .. f'(x)dx = dz; Integration by this last process is evidently equivalent to the integration of a compound function, and is of great importance; for hence it follows that those formulæ of integration which are true for and simple functions of x, are also true for compound functions. SECTION I.-Integration of Fundamental Algebraical Functions. 11.] Integration of x" dx. Let n be substituted for m 1; then m = n + 1; and Therefore to integrate "da, add unity to the index, divide by the index so increased, and by dx. Of this result the following are particular cases: (1.) Let n be negative; that is, for n substitute -n; 12.] The formula (7) is true for all integral and fractional, positive and negative, values of n, with the exception of, n= −1; in which case the right-hand member becomes ∞, and the formula ceases to give an intelligible result: we must therefore return to the principles of definite integration, for they are exact, and by means of them obtain the correct integral. evaluating therefore the indeterminate fraction by the rules of Chapter V, Vol. I; and observing that n is the variable, a result identical with that of Ex. 3, Art. 9; and therefore (8) (9) 13.] Extending the results of Art. 11 and 12 to Compound Functions, as we are authorized to do by Theorem IV, Art. 10, we have [{ƒ (x)}"ƒ'′(x) dx = {f(x)}"+1 (x) dx f(x) n+1 = log {f(x)}. ; (10) (11) Hence the integral of a fraction, whose numerator is the differential of the denominator, is the Napierian logarithm of the denominator. |