what is the integral of cos(log[x])dx where [.] denotes greatest integer function(i.e step value)

∫ cos ( log x ) dx 

integrate by parts 

u = cos (log x) 
du = - sin(log x) dx/x 

dv = dx 
v = x 


∫ cos ( log x ) dx = x cos(log x) + ∫ sin (log x) dx 

again integrate by parts 

u = sin(log x) 
du = cos(log x ) dx/x 

dv = dx 
v = x 

∫ cos ( log x ) dx = x cos(log x) + x sin (log x ) - ∫ cos (log x ) dx 

= 2∫ cos ( log x ) dx = x cos(log x) + x sin (log x ) 

= ∫ cos ( log x ) dx = (1/2)x [ cos(log x) + sin (log x ) ] + C