913 



INTEGRATION, DEFINITE. 



INTEGRATION, DEFINITE. 



914 



As specimens of the reduction of definite integrals, the integrals 



/r-"" 1 cos axt dt /*(-*"'* sin axt tdt 

 r^i t ' 00 ]' 



are severally equal to 



and 



l+P 



[O.oo] 



" dt [-00 , \x- 



/V< a dt [ i* + a, co ] | 



the first having + , the second. Also 



/ ( otl fit / 



[0, oo ] = V ' / ~" * [<f, > ]. 



The following is fundamentally important, 



V* 

 -' cos ozrfa; [0, oo ] = -5 *- Si 



The integral /* *** [0, a] has been tabulated [Diff. Calc., L. U. K., 



J log a; 



p. 662] by Soldner, and a great many integrals may be found from it. 

 Soldner proposes to call it the t'lyarithm-integral of a, and to denote it 

 by the abbreviation li. o. Adopting this notation, we have then, both 

 in definite and indefinite forms, 



/x m dx /*^-*cfcc i. j. 



^*^ ]T a^-M / _^___ = 11. C * 



log a; y a: 



log(o + Jj) 6 



and 10 on. 



Of miscellaneous integrals there is an immense number, of which we 

 give a few instances : 



: [ o,i]=io g r< 1 ;t mr(1 r ) 



T(l+m+n) 



. i 



( 



as cot atie 



(a pos.) /*""" sin ** x~ l dx [0,oo ] =tan ! (6 : n) 



-r- cos ftr rf [0 , = 1 , 

 a; L ' J 2 



if neither a nor a be negative. 



[0, oo]= I V*- 5 * 



1 2ocos 



according as a is less or greater than unity. 



Among the means of producing or using definite integrals which are 

 comprehensive enough to deserve the name of methods, there are four 

 which particularly deserve the attention of elementary writers. 



The first is Laplace's mode of finding the approximate value of a 

 definite integral in which large constant exponents occur. Let <px be a 

 function of x, such as *-'x* or x' (1 - r) m , &c., in which n, m, &c.,are 

 considerable exponents. Let this function vanish when x = a and ;r=4, 

 and. coutinuing positive and finite throughout the interval, let it come 

 to iU maximum Y, when x\. Let v, mean the value of the second 

 differential coefficient of log <px, when x = x, and assume <px=ir-K. 

 Then 



/ $xdx= Y V ( ) / -" dt nearly, 



provided that the limiting values of t on the second side are those 

 which, in the equation of Qx = jf", belong to the limiting values of x 

 taken on the first side. The best approximating cases are as follows : 

 First, when a and b are the limiting values of a-, in which case oo 

 and + oo are those of t, and the result is 



- . V*. 



Secondly, when the limiting values of x are x + {, being small. In 

 this cane 



Secondly, Fourier's theorem, as it is usually called, by which a dis- 

 continuous function can be expressed. This theorem is as follows : 



<px = - ffcos w (x v) . Qv . dwdv, 



from = oo to a:= + oo , and from M = to w=oo . Or thus, the 

 equation 



This method is found, by itself, almost sufficient to meet the wants 

 of the more complicated problems in the theory of probabilities. 

 ARTS A!D 8CI. DtV. VOf,. IV. 



1 /** /*+ 

 = I I cos w (x ?) . #v . 



*"/ / - QO 



e -* 1 " dwdv 



is one which, for all values of x, approaches without limit to truth, as 

 i is diminished without limit positively. But if, instead of the limits 

 oo and + oo , for r, we write a and b, a, being less than 6, then 



i / /cos v>(x i>) . Qvdwdv 

 xj oJ " 



ia a discontinuous function, as follows : From x oo to xa exclu- 

 sive, it is nothing ; when x = a, it is 4o>a ; from x = a to x = b both 

 exclusive, it is <px; when .r=6, it is i<J>6 ; and from x=b to a;=oo it ia 

 nothing. 



Thirdly, the following methods of expanding a function in series of 

 sines and cosines has been extensively used by Lagrange, Poisson, and 

 Fourier. We give it in the most general form after the manner of 

 Poisson. Let 



= AO + 



irx 2ira; 



cos + A, cos -j + . . . 



then, for every value of x from x = to x = i, both inclusive, this 

 equation is true if 



1 /* 2 



Ao = TJ *"** [' 7 ]' A - = 

 Again, the equation 



rx 2rx 



Qx = B, sin -y -t- B g sin -y- + . . . . 



is true from r=0 to x=l, both exclusive, if 



2 / . mvv 



B, = j I sin <f>r<7f[0, q. 



Further, the equation 



** = AO + A, COS H + A ., C( , a "^ + . . . . 



irx 2irx 



+ B t sin -y + B, sin -j + . . . . 



is true for all values of ,r from x= to x= I, both exclusive (becoming 

 fal when a: =0, if 



1 /* 

 2i J 



bo 



= cos 



1 / mire 

 B. = jj sin -j-ffc[0,q. 



But write 2i instead of ?, in the limits only, or write [0, 2?] instead of 

 0, 1], and the equation becomes true for all values of x from to 21, 



th inclusive. 



Fourthly, we shall give two cases of the method deduced by Cauchy, 

 as specimens : the complete method itself has some difficulties which 

 are not yet overcome. 



First, let <px be such a function of x that <f>(x + y \/ \) vanishes 

 when x oo or + oo , whatever y may be, and when y equals oo , 

 whatever x may be. For every root of the form a + A V 1 ( being 

 either positive or negative, and b being positive, but both finite) which 

 makes <f>x infinite, let (x a 6V l)t>z be finite; calculate the value 

 of this last product for each root. For every real root a, of <fu:=ao 

 (x=Q not being one) calculate half the value of (x a) <t>x. Let the 

 sum of all these values and half values be p. Then 



/<pxd.r[-<a, +oo] = 2irV 1 P. 



Secondly, let <px be such a function that <j>(x + y*J 1) vanishes 

 when x= +00 or oo independently of y, and when y= +00 or oo 

 independently of x. Take the imaginary roots only which make Qx 

 infinite, and let (xa b \?l)<t>.r be always finite when + 6^ 1 U 

 one of those roots, and .rr=a + 6V 1. Let the sum of all the values 

 of the last product, for the cases in which A is positive, be P ; and for 

 the cases in which 6 is negative let it be q. Then 



f<t>xdx[ oo,-(-oo]=irv' 1 . (P-Q). 



The subject of definite integrals is one in which the difficulties 

 which have always appeared at the boundaries of mathematical know- 

 ]''il,-i; are condtantly met with. The consequence is, considerable 

 difference of opinion about many pointa. On these, the student who 

 desires to use the higher parts of analysis must hope to form his 

 opinion independently, when his reading and reflection are sufficient 

 for the purpose. Most of these difficulties belong, in principle, to that 



3 N 



