If ` int _(-pi)^(pi) |x sin [x^2-pi}|dx= sum_(r=0)^n a_r sin r ` where [] denote the greatest integer function and `a_r`, `r=0,1,2,...n` are real constants then n is equal to

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

The value of int_(-20pi)^(20 pi) |sin x| [ sin x] dx is (where [.] denotes greatest integer function)

int_(0)^(2pi)[|sin x|+|cos x|]dx , where [.] denotes the greatest integer function, is equal to :

int_(0)^(2pi)[|sin x|+|cos x|]dx , where [.] denotes the greatest integer function, is equal to :

cos ^(3) x sin 2x = sum _(x =0) ^(n) ar sin (rx) AA x in R, then

If the definite integral I=int_(0)^(pi)sin[x] dx=sum_(epsilon=0)^(n)a_(epsilon)sin epsilon . (where, [.] is the greatest integer function), then the vlaue of (a_(n)+n)/(pi) is equal to

If the definite integral I=int_(0)^(pi)sin[x] dx=sum_(epsilon=0)^(n)a_(epsilon)sin epsilon . (where, [.] is the greatest integer function), then the vlaue of (a_(n)+n)/(pi) is equal to

If R = (sqrt(2) + 1)^(2n+1) and f = R - [R] , where [ ] denote the greatest integer function, then [R] equal

If k in N and I_(R)=int_(-2k pi)^(2k pi)|sin x|[sin x]dx where bl denotes the greatest integer function,then sum_(k=1)^(100)I_(R) equals

If f : R to [-1, 1] , where f(x)=sin(pi/2[x]) (where [.] denotes the greatest integer function), then the range of f(x) is