If `I_(m, n) = int_0 ^(pi/2) sin^mx cos^nx dx` then show that `I_(m, n) = (m-1)/(m+n) I_(m-2, n)` and find `I_(m, n)` in terms of different combinations of m and n.

If I_(m"," n)=int cos^(m)x*cos nx dx , show that (m+n)I_(m","n)=cos^(m)x*sin nx+m I_((m-1","n-1))

If I_(m,n)= int(sinx)^(m)(cosx)^(n) dx then prove that I_(m,n) = ((sinx)^(m+1)(cosx)^(n-1))/(m+n) +(n-1)/(m+n). I_(m,n-2)

If I(m)=int_(0)^( pi)ln(1-2m cos x+m^(2))dx then I(1)=

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_m-2n(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4w h e nbot hma n dna r ee v e n((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))

int sin mx cos nxdx,m!=n

If I(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx, then

int_(0)^( pi)cos mx*cos nxdx(m!=n)=