Let `R=(5sqrt(5)+11)^(2n+1)a n df=R-[R]w h e r e[]` denotes the greatest integer function, prove that `Rf=4^(2n+1)`

If p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot] denotes the greatest integer function, then the value of p(1-f) is equal to a. 1 b. 2 c. 2^n d. 2^(2n)

Evaluate: int_0^(2pi)[sinx]dx ,w h e r e[x] denotes the greatest integer function.

Evaluate: int_(-5)^5x^2[x+1/2]dx(w h e r e[dot] denotes the greatest integer function).

Evaluate: int_1^(e^6)[(logx)/3]dx ,w h e r e[dot] denotes the greatest integer function.

Prove that int_0^oo[cot^(-1)x]dx ,w h e r e[dot] denotes the greatest integer function.

Evaluate: int_0^2[x^2-x+1]dx ,w h e r e[dot] denotos the greatest integer function.

Statement 1: If p is a prime number (p!=2), then [(2+sqrt(5))^p]-2^(p+1) is always divisible by p(w h e r e[dot] denotes the greatest integer function). Statement 2: if n prime, then ^n C_1,^n C_2,^n C_2 ,^n C_(n-1) must be divisible by ndot

f:(2,3)vec(0,1)d efin e db yf(x)=x-[x],w h e r e[dot] represents the greatest integer function.

Evaluate lim_(n->oo) [sum_(r=1)^n(1/2) ^r] , where [.] denotes the greatest integer function.

Evaluate: int_0^(10pi)[tan^(-1)x]dx ,w h e r e[x] represents greatest integer function.