Show that: `int_0^[[x]] (x-[x])dx=[[x]]/2`, where `[x]` denotes the integral part of `x`.

Evaluate: int_-100^100 Sgn(x-[x])dx , where [x] denotes the integral part of x .

Show that: (int_0^[x] [x]dx/(int_0^[x] {x}dx)=[x]-1 , where [x] denotes the integral part of x and {x}=x-[x] .

Prove that int_0^x[t]dt=([x]([x]-1))/2+[x](x-[x]), where [.] denotes the greatest integer function.

If int_-1^1(sin^-1[x^2+1/2]+cos^-1[x^2-1/2])dx=kpi where [x] denotes the integral part of x , then the value of k is …

if f(x) ={x^(2)} , where {x} denotes the fractional part of x , then

int_0^1 [x^2-x+1]dx , where [x] denotes the integral part of x , is (A) 1 (B) 0 (C) 2 (D) none of these

int_0^2 x^3[1+cos((pix)/2)]dx , where [x] denotes the integral part of x , is equal to (A) 1/2 (B) 1/4 (C) 0 (D) none of these

Evaluate int_(0)^(2){x} d x , where {x} denotes the fractional part of x.

Statement-1 : lim_(x to 0-) ("sin"[x])/([x]) = "sin" [x] != 0 , Where [x] is the integral part of x. Statement-2 : lim_(x to 0+) ("sin"[x])/([x]) != 0 , where [x] the integral part of x.

the value of int_(0)^([x]) dx (where , [.] denotes the greatest integer function)