PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 585 



sin (n + m) tt 



also /?* 4- m + 1 = {n + m) (n + m — 1) ...(« + 1) /« + 1 



d X-'" ~ (w + 1) (n + 2)T. . (n + ni) 



Ohs. The introduction of arbitrary functions (of integration) 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. 



d^ x^ 

 Ex. 4. Find the value of ^^. • 



Here we must use the second formula ; and we get 



^ = {-if_[±_ __^±JL^ (^»-/^_««-^) 



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



Now, —^—, ^ — = — loar — • 



saxin — fj) TT TT *= a 



TT ° a 



= ^-i "^ ^* log " 



IT "a 



But /i =\/7r by the well known formula /r /1-r = ^^,^7^ ; putting >'=i 



d^xi ^/_l 1 



« 



= ^-. — log., 



if we omit the constant. 



d X 

 Ex. 5. Find the value of 1- 



dx - 



f/i T^ /I- sin I- TT 



Byformula(2) ^^^ = (-1)' 4^ ' sin^ " - 



Ex. 6. Find the value of 



= 00 



<^2 ^ 2 



c?a; 



