Please solve the following integral:
Undetermined error: @@lim(int((kx-box{kx})^k dx,0,kbox{x}),k,infty)@@ ; k belongs to N (where box(.) denotes grestest integer function)
NOTE:- Please consider box(.) as [.] for convenience. Since the system wasn't accepting [.] symbol, I wrote it as box(.)
(a) box(kx) (b) box(x) (c) box(Undetermined error: @@x/k@@) (d) box(Undetermined error: @@x^k@@)
The ans. is given as b. Please reply.
I am not getting the answer but may be we can proceed like this:
We know that x-[x] i.e. the fractional part of any number say x lies between 0 and 1.Therefore, kx-[kx] also lies between 0 and 1.
0< kx-[kx]<1
0< {kx-[kx]}k<1k
Therefore the integral of {kx-[kx]}k where x is a variable ,lies between 0 and x.Putting proper limits ,we find that the given integral lies between 0 and k[x]. Now put k tending to infinity.You will find that the integral lies o to infinity.
IF x does not lie between 0 and1 then integral is smaller than k[x].k is a natural number and can take large values ,[x] is an integer .Now when I multiply an integer with another integer then I am again going to get the product as an integer which can be written as another [x].Hence the integral is smaller than [x]. I do not know whether the equality sign holds good or not.Somebody please reply whether the given solution seems to be correct or not.