Plainly, exchanging extremes (a and b) merely reverses the integral by reversing f to a a b I a dx, f b a = . Alsof = + f e ,f(x) being supposedly integrable throughout, from a to b, and to c. Also fc0x.dx=cf0(x)dx. Also f b b ±e f(x)dx= f f (xzt.-.c)d(x-i-c),a mere change a*c of origin. Also f [0(x) ±_-0(x)]clx= f 0(x)dx :f.-_ 0(x)dx. We easily prove the almost obvious f Theorem of the Mean: If f(x) =-.(1)(x) 0(x), each factor integrable, and 0(x) always of same sign, in [a, IA, then 0(x) 0(x) dx = b 0(70 i Q(x)dx, xTE a +0(b a). Hence /6 .-__. a f(b), la= f (a), where the D is again the front of variation of the Integral I or Area A.
Hence we readily reckon many integrals as: f 6 xfsdx= an-I-1 xis+1 n a pos. int.; o n+1 n+1 n+1 a b b icosxdx---- {sin x 1 a f , a sin x dx=--. { cos x a , b I 1 X b 0 V V + a This is seen at once from the figure (circle quadrant), as also f: V 0 + = I{ x V i-xa + r seen from We perceive that the f(x) is always the D of the so-called Indefinite Integral, the expression to be evaluated at b and a. This is easily proved variously to be always the case. Thus, if f(x)=0'(x) for x in [a, ], then 11 Q(b)tb = f(b) #'(b) = 0, for every value of b in the range of Integra bility. Hence I-0 (b)=C. For b=a, C is found to be = 0(a) ; hence, (X)(IIC == 0(b) 0(a). a Hence, to calculate the integral of any integrand from a to b, find the function of which the integrand is the D, and take the difference of its values at b and a. The D of the integral is the integrand, so far as form goes, but the value depends on the extremes. Since b may be any x in the range of integra bility, it is common to write it x, using x in double sense, not necessarily confusing. So long as a is unassigned, C is undetermined; hence it is common to omit a and write = 0(x) + C, where under f we may put z or any other symbol for x. The integral depends for its form solely on f; for its value, on a and b also.
Hence integration and derivation or differen tiation are inverse operations i i D=-- I . The direct D yields a definite result, the inverse J yields a result definite only as to form, up to an additive constant, C. (Cf. Evolute and Involute, above). Derivation simplifies; reducing even transcendents to algebraics; Integration complicates, lifting algebraics and even rationale up into tran 1 scendents (as V and . Derivation is
deductive and can create no new forms. Integration is inductive and creates an 00 of new forms, all defined as integrals.
Operating directly on y=f(x) by a series of differentiations as to x, say 0(D), we get some function of x, as X, or #(D))=X. If we know X and 0, we may seek that (x) that will yield X on being subjected to the train of operations 0(D) , i.e., we seek to invert at once the totality of operations 0(D), X so that y= 0( (D)X. ' This inversion is solving the Differential Equation 0(D)3=X, and is perhaps the most profound of mathe matical operations, of immense and even uncon querable difficulty, overcome as yet only in d'Y special cases. Thus x' =x dy dx + 4y = where #(D)m x'D'--3xD-1-4, yields, as result 1 of the inverse or 0(D)' log x+log where A and B are arbitrary constants. Other forms of 0(D), quite as simple, yield far higher transcendents.
Inverting a table of elementary D's we get a table of elementary Integrals. The art is to reduce other forms, if possible, to these elementary forms; when impossible, we must introduce transcendents defined by integrals.
Change of Variable is the most fruitful method of Reduction. By mediate deriva tion, = 0'(u) also Du0(u) -= 0'(u); hence fr(u)du=0(si)--40'(u) dx. In this 2d5, all is supposed expressed through x. Hence, to pass from an old to a new variable of integration, multiply by the D of the old as to the new, or divide by the D of the new as to the old; i.e., under thei sign, du dx, dx --= .
Of course, the extremes must be properly adjusted.
Integration by parts is also a powerful reduc tive process. From +rsvz we have Iry =Jiro& This latter 5 may be simpler, or may return into the first, or other advantages may accrue.
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ThUSPOS x . cos x dx=sin x cos x ism x dx = sin x cos x + x i + cos x whence 2icos dx=-x +sin x cos x. Similarly for any integral power of sine or cosine except (sin x)-', (cos x) -', which are reduced by x passing to the half-angle, T.
What is the range of such reductions? What functions can we thus integrate in terms of known functions? Few enough. Of Alge br'aics, I. Rational functions, (x by decomposi 0 (x) ) ' fion into part-fractions.