If `f(x)={0` ,where `x=n/(n+1),n=1,2,3.... and 1` else where then the value of `int_0^2 f(x) dx` is

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

For each ositive integer n, define a function f _(n) on [0,1] as follows: f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):} Then the value of lim _(x to oo)int _(0)^(1) f_(n) (x) dx is:

If f(x) is a polynomial of degree n such that f(0)=0 , f(x)=1/2,....,f(n)=n/(n+1) , ,then the value of f (n+ 1) is

If f(x) = n , where n is an integer such that n le x lt n +1 , 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

If f(alpha)=f(beta) and n in N , then the value of int_alpha^beta (g(f(x)))^n g\'(f(x))*f\'(x)dx= (A) 1 (B) 0 (C) (beta^(n+1)-alpha^(n+1))/(n+1) (D) none of these

Suppose for every integer n, .int_(n)^(n+1) f(x)dx = n^(2) . The value of int_(-2)^(4) f(x)dx is :

If f(x)={{:(,x([(1)/(x)]+[(2)/(x)]+.....+[(n)/(x)]),x ne 0),(,k,x=0):} and n in N . Then the value of k for which f(x) is continuous at x=0 is

The value of integral sum _(k=1)^(n) int _(0)^(1) f(k - 1+x) dx is

If int_0^oo x^(2n+1)\ e^(-x^2)\ dx=360, then the value of n is___