`L e t \ f(n)=1+1/2+1/3+...+1/n`. Then show that `f(n)=int_0^(pi/2)cot(theta/2)(1-cos^ntheta)d theta`

Prove that: (sin2theta)/(1+cos2theta)=t a ntheta

Prove that: (sin2theta)/(1+cos2theta)=t a ntheta

The value of I(n) =int_0^pi(sin ^2n theta)/(sin^2theta) d theta is (AA n in N)

If f(x)=int_0^(pi/2)ln(1+xsin^2theta)/(sin^2theta) d theta, x >= 0 then :

If mgt0, ngt0 , the definite integral I=int_0^1 x^(m-1)(1-x)^(n-1)dx depends upon the values of m and n is denoted by beta(m,n) , called the beta function.Obviously, beta(n,m)=beta(m,n) .Now answer the question:The integral int_0^(pi/2) cos^(2m)theta sin^(2n)theta d theta= (A) 1/2beta(m+1/2,n+1/2) (B) 2beta(2m,2n) (C) beta(2m+1,2n+1) (D) none of these

Let n ge 2 be a natural number and 0 lt theta lt (pi)/(2) , Then, int ((sin^(n)theta - sin theta)^(1/n) cos theta)/(sin^(n+1) theta)d theta is equal to (where C is a constant of integration)

Show that tan{(2n+1)pi+theta}+tan{(2n+1)pi-theta}=0

Iflambda=int_0^(pi/2)costhetaf(sintheta+cos^2theta)dtheta a n dI_2=int_0^(pi//2)sin2thetaf(sintheta+cos^2theta)dtheta,t h e n I_1=-2I_2 (b) I_1=I_2 2I_1=I_2 (d) I_1=-I_2

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

Solve the equation sin^(2)ntheta - sin^(2)(n-1)theta = sin^(2)theta