PROFESSOR KELLAND ON GENERAL DIFFERENTIATION, 

 sin » TT ., 



585 



(-1)" 



sin (» + m)'W 



also 



/ii + m+1 = (ri + m) (n + m — l)...(n + Y) In + 1 



dx-"" (w+ 1) (n + 2) ... (w.+ m) ■ 



Obs. The introduction of arbitrary functions (of integi'ation) is of course re- 

 quisite to render this complete ; but we shall defer the discussion of the nature 

 of these functions to a separate section. 



Ex. 4. Find the value of 



dx'^ 



Here we must use the second formula ; and we get 



d'-x'' 



dx 



^ = (-l)^il 



sin ^ TT 



(^^ 



7! — II. _,,«■— li.\ 



n sin (»—/z)7r 

 if we suppose the differential to vanish when x=a. 



Now, 



sin (n—jJL) IT IT 



log- 



TT ° a 



dxi 



s/^ 



IT 



log 



But / ^ =\/Tr by the well known formula /r /1-r 

 dixi ^/Z^i 1 



dx^ 



2 ^^^°ST 



or 



if we omit the constant. 



Ex. 5. Find the value of 



^/=T 



n/TT 



logx, 



d^ x'- 



dx 



i 



By formula (2) 



dx*- [2 



Ex. 6. Find the value of 



= 00 . 



d'ix*- 

 dx^ 



sm |- TT 

 sinTT 



putting r=^\ 



