If `[x]` denotes the integral part of `[x]`, show that: `int_0^((2n-1)pi) [sinx]dx=(1-n)pi,n in N`

If [x] denotes the integral part of x and a_(n)=sum_(r=0)^(n-1)[x+(r)/(n)] then find Lt_(n)rarr oo(a_(1)+a_(2)+...a_(n))/(n^(2))

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) = int_(0)^(pi)(sinx)^(n) dx, n in N then

If n is a positive integer, prove that: int_0^(2pi) (cos(n-1)x-cosnx)/(1-cosx)dx=2pi , hence or otherwise, show that int_0^(2pi) (sin((nx)/2)/sin(x/2))^2dx=2npi .

Let I_(n)=int_(0)^(pi//2) sin^(n)x dx, nin N . Then

int_(0)^(x)[sin t]dt, where x in(2n pi,(2n+1)pi),n in N, and [.] denotes the greatest integer function is equal to -n pi(b)-(n+1)pi2n pi(d)-(2n+1)pi

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

The value of the integral int_(0)^(1) x(1-x)^(n)dx , is