If `U_n=int_0^(pi/2)(sin^2n x)/(sin^2x)dx,` then show that `U_1,U_2,U_3.......U_n` constitute an AP. Hence or otherwise find the value of `U_n.`

If U_n=(sqrt(3)+1)^(2n)+(sqrt(3)-1)^(2n) , then prove that U_(n+1)=8U_n-4U_(n-1)dot

If U_n=int_0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zero, then show that U_(n+2)+U_n=2U_(n+1)dot Hence, deduce that int_0^(pi/2)(sin^2ntheta)/(sin^2theta)=1/2npidot

If U_(n)=int_(0)^(pi)(1-cosnx)/(1-cosx)dx where n is positive integer of zero, then The value of U_(n) is a. pi//2 b. pi c. npi//2 d. npi

u=int_0^(pi/2)cos((2pi)/3sin^2x)dx and v=int_0^(pi/2) cos(pi/3 sinx) dx

If A= ((1,0,0),(2,1,0),(3,2,1)), U_(1), U_(2), and U_(3) are column matrices satisfying AU_(1) =((1),(0),(0)), AU_(2) = ((2),(3),(0))and AU_(3) = ((2),(3),(1)) and U is 3xx3 matrix when columns are U_(1), U_(2), U_(3) then answer the following questions The value of (3 2 0) U((3),(2),(0)) is

If u+t=5 and u-t=2 , what is the value of (u-t)(u^2-t^2) ?

int_(1)^(2)((x^(2)-1)dx)/(x^(3).sqrt(2x^(4)-2x^(2)+1))=(u)/(v) where u and v are in their lowest form. Find the value of ((1000)u)/(v) .

f(x)={(2xtanx-pi/(cosx) ,, x!=pi/2),(k ,, x=pi/2):} i s \ c o n t i n u o u s \ a t \ x=pi/2, then find the value of k .

A stone is thrown horizontally with velocity u. The velocity of the stone 0.5 s later is 3u/2. The value of u is

If U_n=int_0^1x^n(2-x)^ndxa n dV_n=int_0^1x^n(1-x)^n dx ,n in N , and if (V_n)/(U_n)=1024 , then the value of n is________