i need some help or understanding or a book or material that explains this type of problem
find the value of
That is an example of a "Stieljes" integral. The original "Riemann" integral, , divides the x-axis into sub-intervals, , and then measures the "area" by where is just the length, . The "Stieljes" integral, , generalizes that by measuring each sub-interval by where can be any increasing function of x. If happens to be differentiable the Stieljes integral reduces to a Riemann integral: .
But does not have to be differentiable or even continuous. One application of the the Stieljes integral is to be able to write a sum as an integral: if then is non-zero only for and to be on different sides of an integer: .
That, together with the fact that
should be enough for you to do this problem.
(By the way, I have interpreted your "[x]" as the floor function. Latex has "lfloor" and "rfloor" to get .