If `m!=n` and `(m+n)^(-1)(m^(-1)+n^(-1))=m^xn^y` then show that `x+y+2=0`

If m =! n and (m + n)^(-1) (m^(-1) + n^(-1)) = m^(x) n^(y) , show that : x + y + 2 = 0.

If a^(x)=m,a^(y)=n and a^(2)=(m^(y)n^(x))^(z) show that xyz=1

If x^(m).y^(n)= (x+ y)^(m+n) then show that, (d^(2)y)/(dx^(2))=0

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-x)^(n-1)dx . It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0 . int_(0)^(1)(x^(m-1)+x^(n-1))/((1+x)^(m+n))dx=

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx . It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0. int_(0)^(oo)(x^(m-1))/((1+x)^(m+n))dx=

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-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-2,n)(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/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))}