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

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

Determine a positive integer n such that int_0^(pi/2)x^nsinx dx=3/4(pi^2-8)

Determine a positive integer n such that int_0^(pi/2)x^nsinx dx=3/4(pi^2-8)

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 .

Prove that the value of : int_0^pi(sin(n+1/2)x)/sin(x/2)dx=pi

The domain of the function f(x)=1/(sqrt({sinx}+{sin(pi+x)})) where {dot} denotes the fractional part, is (a) [0,pi] (b) (2n+1)pi/2, n in Z (c) (0,pi) (d) none of these

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

int_0^x[sint]dt ,w h e r ex in (2npi,(2n+1)pi),n in N ,a n d[dot] denotes the greatest integer function is equal to -npi (b) -(n+1)pi 2npi (d) -(2n+1)pi

Compute the integrals: int_0^oof(x^n+x^(-n))logx(dx)/(1+x^2)