INTEGRATION, DEFINITE. 



INTEGRATION, DEFINITE 



will easily be seen that c u '+ u is a solution for any value of c and x, 

 prorided only that n=atf: and alo that the mm at any number 

 of such terms u a solution. Hence we assume an indefinite number 

 ef such terms. giving to c the form +*.d, and summing them with 

 such value* of r as will make the whole repreaent 



and we than tea that this integral U a solution or general value of u, 

 whatever the function f may be, and whatever may be the values of 

 f and 5. By a reduction which i rendered easy by some of the 

 reeults presenUy mentioned, this solution is thrown into the form 



where if may be the symbol of any function. From this it is clear 

 that the given differential equation has numberless solutions which 

 ordinary symbols are incapable of expressing in finite terms. The 

 treatise in the Library of Useful Knowledge on the Differential 

 Calculus, Gregory's ' Examples of the Differential Calculus,' and the 

 Cambridge Mathematical Journal,' and its successor, contain various 

 examples of this mode of expression applied to differential equations. 



We now proceed to give a selection from the enormous number of 

 definite integrals which has been given. They have been found by 

 detached methods, so that we could not attempt to give anything more 

 than the results. Our article U intended for reference to the forms 

 which it is probable will be noted in future elementary works, and 

 which the mathematical reader may also wish to refer to. In order to 

 avoid risk of broken or dropped letters, in on article in which the 

 correct printing of the limits is of the utmost importance, we shall 

 print what is usually denoted by_/^ juxU in the following way, 

 /Qxdx [a, 6]. Any conditions as to the values of constants will be 

 ejiumsul before the integral. It need hardly be said that the article 

 FACTORIALS must be considered as a part of the present one. 



Among the integrals which clearly depend on, or are connected with, 

 factorials, are the following : 



x~dx 



(m and n positive) f * ' (1 - *)-' dx [0, 1] = r(m + n ) 



/m in *)*~' Tm . I*ii 



(x + a)"'*'" "- ' " = o(l + o)" r(w + fi) 



(n positive) y< -log *)- dx [0, 1] = Fn 



(M and poritivejy*-- ' ( -log a:)*-' dx [0, 1] = - P 



(a and it positive^*"-' t~"dx [0, eo]=Tii 



j j 

 ( positive) J- f dx [0, oo] = r - 



Tables of the value of (2 : \/t) J"f~ a dx [0, a], which are of great 

 importance in the theory of probabilities, are given in modern works 

 on that subject. The following expression by means of a continued 

 fraction is useful. Let j = 1 : 2a : , then 



y being as in FACTORIALS. 



One of Killer's integrals, generally called the second Eulerian 

 integral, the factorial integral being the first, he denoted by the symbol 



- ) ; it is /-> (!-)?- <tt [0, 1], 



and it U included in those already given. 



There is a class of multiple integrals closely connected with factorials, 

 which may be made to savo much trouble in applications to geometry. 

 W* shall take three variables as a specimen, but the same formula; 

 may be written with any number. The triple integration being made 

 for all punitive values which give x + y + z not exceeding I, we have 

 (a, b, e, being positive) 



ra.rb.re . 



. 



Similarly, the condition being that 



shall not exceed I, we have 



We shall now give some specimens of the results of functions 

 involving trigonometrical quantities. One of the most important of 

 this class is the following : 



y*-' siu bx dx [0, 



according as o is positive or negative. 



/cos ox dx 



= 4 



.xdx 



/"e cos&ntr |0, 



, oo]=6 :( 



from these come 



/ cos bxd.t [0, ] = 0, / sin bxdx [0, oo ] = 1 ; 



and from these come two equations which have been much used, long 

 before they were openly expressed, 



sin oo = 0, cos oo =0. 



Some difference of opinion exists about these equations, which in 

 fact involve a great deal of what has been done by mathematicians * in 

 the last thirty years. 



When a and n are both positive 



TB.cos{n tan-' (6:0)} 

 /- cos 6x . a--' <tc [0. w]= - 



' 



ft-" rin bx . 



, o ] 



/cos *" . vdx [0, OB ] = 5 



1 

 'sin *- . x*dx [0, oo ] = - 



dx 



But when a=2mbc, m being an integer, the preceding integral is 



2 *-- ' 



This is a specimen of a sort of discontinuity which very frequently 

 occurs, and from not attending to which mistakes have often arisen. 



If we call 4( + f-) and 4 (-%-) the hyperbolic cosine and sine 

 of x, and denote them by h. cos .c and h. sin .c, we have, the limits 

 being and oo , and a being less than *, 



'h. nin ax 



sin a 



sin ex dx = 



2 cos a + h. cose 



cos fa . h. cos \c 



cos a+h. cose 



1 b. sine 



2 cosa + h. cose 



. sin ja . h. sin je 



sm ** co a + h. cos c 



1 +1 1 

 [.]= 4 ?^1 ~ & 



/P^- t dt 

 jm_l [0, ] i the th number of Bernoulli 



[NUMBERS or BERNOULLI], meaning that opposite to which 2 - 1 is 

 written in the article cited. 



With reipml to thne equations, it mut b observed that they are not to 

 bare their tlgobraical conncqui nee. ; thiu, tln'o i> not 0, but j. The truth 

 Menu to be,a> far u vet appears, that any function *. which becomes ind8nit 

 in form, by ths angle * becoming infinite, U properly represented by / fid* 

 f 0,2T] divided by 2. 



