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int(-2)^(4)[x]dx," where "[x]" is integr...

int_(-2)^(4)[x]dx," where "[x]" is integral part of "x

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int_(-2)^(4) f[x]dx , where [x] is integral part of x.

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

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

Evaluate : (i) int_(-1)^(2){2x}dx (where function {*} denotes fractional part function) (ii) int_(0)^(10x)(|sinx|+|cosx|) dx (iii) (int_(0)^(n)[x]dx)/(int_(0)^(n)[x]dx) where [x] and {x} are integral and fractional parts of the x and n in N (iv) int_(0)^(210)(|sinx|-[|(sinx)/(2)|])dx (where [] denotes the greatest integer function and n in 1 )

Show that: int_0^x[x]dx=[x]([x]-1)/2+[x](x-[x]) , where [x] denotes the integral part of x .

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

Show that: int_0^[[x]] (x-[x])dx=[[x]]/2 , 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] .

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] .