If m, n in N , then int_(0)^(pi//2)((sin^(m)x)^(1/n))/((sin^(m)x)^(1/n)+(cos^(m)x)^(1/n))dx is equal to

Comprehension 1 Let I_(n,m)=intsin^(n)xcos^(m)x.dx . Then we can relate I_(n,m) with each of the following i) I_(n-2),m , ii) I_(n+2),m , iii) I_(n,m-2) iv) I_(n,m-2) , v) I_(n-2,m+2) , vi) I_(n+2,m-2) Suppose we want to establish a relation between I_(n,m) and I_(n,m-2) , then we set P(x)=sin^(n+1)xcos^(m-1)x ................(1) In I_(n,m) and I_(n,m-2) the exponent of cosx is m and m-2+1=m-1 . Now choose the exponent m-1 of cosx in P(x). Similarly choose hte exponent of sinx for P(x). Now, differentiating both sides of (1), we get P^(')(x) = (n+1)sin^(n)xcos^(m)X-(m-1)sin^(n+2)Xcos^(m-2)X =(n+1)sin^(n)Xcos^(m)X-(m-1)sin^(n)x(1-cos^(2)x)cos^(m-2)X =(n+1)sin^(n)X cos^(m)X-(m-1)sin^(n)Xcos^(m-2)X+(m-1)sin^(n)Xcos^(m)X =(n+m)sin^(n)Xcos^(m)X-(m-1)sin^(n)Xcos^(m-2)X Now, integrating both sides, we get sin^(n+1)cos^(m-1)x=(n+m)I_(n,m)-(m-1)I_(n,m+2) Similarly, we can establish the other relations. The relation between I_(4,2) and I_(2,2) is