Let `f(x)=(sin^(101)x)/([(x)/(pi)]+(1)/(2))`, where [x] denotes the integral part of x is

Statement-1: Let f(x)=[cos x+sin x),0

If f:(0,pi)rarr R and is given by f(x)=sum_(k=1)^(n)[1+sin((kx)/(n))] where [] denotes the integral part of x, then the range of f(x) is

Let f(n)=[(1)/(2)+(n)/(100)] where [x] denote the integral part of x. Then the value of sum_(n=1)^(100)f(n) is

Let f(x)=(x(sinx+tanx))/([(x+pi)/(pi)]-1//2) (where (.] denotes the greatest integer function) then find f"(0) .

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

Let f(x)=[x] +sqrt({x}) , where [.] denotes the integral part of x and {x} denotes the fractional part of x. Then f^(-1)(x) is

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

The functions f and g are given by f(x)=(x) the fractional part of x and g(x)=(1)/(2)sin[x]pi where Ix] denotes the integral part of x, then range of gof(x) is