B0 



INTEGRATION. 



INTEGRATION, DEFINITE. 



910 



write m 1 for m and we have, by transformation, 



d>(m 1, x) 1 



v " = ~ Mm- 1) + it(m 1) V 



The two first formulas become (p being laxx-) 



(Jit m + 1) ( m, n) 2a ( m 1, n 1] 



(n m + 1) ( m,n) ( m 2, n l) = .c-"+'p 

 from which, writing v,,,. for ( m, n) or/Jtr^Fdz, we have 

 1 (Zax x?)*, 2na 



* 2--t-l v "- 1 '- 

 



v "." = n m + 1 a"- 1 



- m + 1 



If it were required to reduce n in the preceding without altering m, 

 throw the formula ar-(2*r *)*, into the form x*-"(2ax)* and 

 use the first of the four formulae. 



29. All the preceding forms involving x"t* are particularly in use 

 when n is a fraction, positive or negative, with the denominator 2. 

 These in fact form the most usual cases. Fonnulse involving the 

 powers of A + BZ + oz 1 ' are so little wanted, that they are better omitted 

 in a work in which space is of importance. 



30. Let s and c stand for gin and cos 0. The integral /a" c"<0, m 

 and n being positive integers, can be immediately found if s'c" be 

 reduced to the sum of terms of the form A cos kt or A sin M. But 

 this process is laborious, except when a simple rule is mastered which 

 seldom appears in works on trigonometry, and is seldom used except 

 when there is some particular reason for exhibiting the result in the 

 form of simple sines and cosines. 



31. The following equations of reduction are those which are most 

 commonly used : 



/_ eg**-' m 1 /* 

 * / 



/sc*-' 1 / 

 J 



/de c m 2 /* de 



? = ~ (m-l)a ' + m-l / s-- 

 /^? " 2 / rf9 



c 1 " = (-l)c"-' + n-1 / c " 



ys"c"d = 



s"c 3 rf0 



/e'c?9 _ n-1 /*c^*d 



~ (m ljs"~ l m 1 / s*~* 

 /*s"rf9 s*~ * m 1 /'s*~ 5 cW 



m + n 2 



* -jrn- 



re + n-2 /* rftf 

 m-1 y s -c" 



/c'rfg -l /c'-'d* 



(n-mjs"-' n m / s" 

 c*" 1 "' m42 ftfde 



(m-l)a ' " m-l / iF 1 * 



32. We have given the last steps in various forms, because in fact 

 all the integrals of the form /r"(o' :i?)*dx depend upon them. 

 For if x = a sin t, the last integral becomes o" + * + '/sin " 9 cos* 1 + ' M9. 



We have now given most of the forms which will be useful in an 

 ordinary work of reference. Further forma and examples will be 

 found in many works on the integral calculus, but the largest collec- 

 tion ut in Meier Hirsch's ' Integraltafeln,' Berlin, 1810, 4to, a work oi 

 which there is lo an Knglish edition. 



We hve omitted notice of a great many such forms fx*t"dr 

 /c*t~ co nxdx, Ac., which are little used, except in particular cases 

 When fx . "cfa can be integrated, it follows that QX . "cos bxdx. 

 &c., can alo be integrated, since the second can be made into the sum 



/i'dC *-' m-l /s"-^rffl 



"c"~ ~ (m-n)c 1 + m-n ./ ~~c~ 



a"+' m-+2 /-"rfa 



= (-l)c"-' " ' n-1 J c"- 



/tan"- 1 9 / 

 tan"9rf= n-1 - - / t 



or difference of two functions of tha first form, by putting for cos tx 

 or sin bx their exponential values. 



The question of the possibility of integration in finite terms call 

 often be settled by the following theorem : Integration and dif- 

 'erentiation, with respect to different variables, are convertible 

 operations; thus 



dfudx /'du 



j = / j~ dx. 

 dy J ay 



If therefore fudx can be found, so also can f(du, : dy)dx, if y be not 

 function of x. From this it will be seen that whenever <t>xe*dx can 

 be integrated so can <t>Xf"x*dx, which ia obtained by n differentiations 

 with respect to a ; and also that whenever <f>x.x*dx can be integrated, 

 so can <t>xx* (log x) m dx, which is obtained by m differentiations with 

 respect to n. 



Functions involving the transcendental forms sin ~ l <t>x, &.., can 

 sometimes be reduced to more algebraical forma by integration by 

 parts. Thus, 



//* x'/vrfj; 

 v sin-' x . dx = sin-' X ^J\dx - J y ^ _ ^ ax 



//x'/vda; 

 v log srfz=log x .fvdx - J - ax, &e. 



in which x' means <?I : dx. 



INTEGRATION DEFINITE. In the preceding article we have 

 given some idea of the usual modes of integration. The results, 

 which in the present article are given under the name of definite integrals, 

 are mostly cases in which it is possible to find an integral when both 

 limits are given [INTEGRAL CALCULUS]; but not possible to find the 

 integral in all cases. If we can integrate tyxdx generally, that is, if we 

 can find the function ,x, of which <fix is the differential coefficient, 

 we can always express the integral, the limit of the summation in the 

 article just referred to, aa follows : 



but it frequently happens that Qx is a function for which this cannot 

 be done in a finite form, except for certain values of a and 6. And it 

 happens almost as frequently that these practical values are of 

 particular importance. 



But the view of definite integrals which best shows their utility is 

 the consideration of them as fundamental modes of expression. The 

 ordinary symbols of algebra, it is well known, are incompetent to 

 express in finite terms by far the greater number of integrals. Con- 

 sequently the integrals themselves become modes of expression, and 

 frequently the only ones. When we find a language with which we 

 have much to do, and which ha words which cannot be translated, we 

 adopt the words of that language into our own. Precisely the same 

 thing is done in the case of definite integrals. Thus, in FACTORIALS, 

 we adopt the integral fie'x'dx, as the fundamental mod? of expres- 

 sion for a function tiU then inexpressible, which becomes ] 2 . 3 . . . n 

 whenever n is an integer, and remains intelligible, though not very 

 easily found, when is a fraction. 



Further to illustrate this, let us suppose that the integral calculus 

 had made some progress before the conception of a logarithm had been 

 formed : a thing which might easily have happened. It would then 

 have been found that fx~ l dx was wholly unattainable, a function which 

 algebra could not express in finite terms. It would therefore itself 

 have become a mode of expression, and it would soon have been, 

 proved that 



/'dx i tt dx t"*dx 

 l x + J i * m j i x' 



Here then would have been an obvious indication of tho existence of a 

 function proper to be made use of in performing multiplication by 



means of addition, &c. ; and tables of the values of J.x~ l dx would have 

 been formed by the method of quadratures [QUADRATURES,] or other- 

 wise ; which would, so it happens, have been a much easier task than 

 that which fell on the first calculators of logarithms. For all this 

 however it happens that we are prepared by knowing logarithms and 



their properties ; so that_/i~' dx is seen to be log. x + c, and J^ x dx to 

 be log. a : the logarithms throughout this article being Naperian. 

 But we are not equally ready forfe dt, nor for A x' dx (except 



when is integer) nor for y cos x dx : and accordingly we are obliged 

 to study the properties of these functions as fundamental modes of 

 expression. 



To give some idea of the use of this view, we exlubi a mode of 

 solving the following partial differential equation, 



if " d^' 



the general solution of which cannot be expressed in finite terma. It 



