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

Comprehension 1 Let I_(n,m)=intsin^(n)xcos^(m)x.dx. Then we can relat

1.	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
Answer» `I_(4,2)=1/6 (-SIN^(3)xcos^(3)X+3I_(2,2))` `I_(4,2)=1/6(sin^(3)x COS^(3)x-3I_(2,2))` `I_(4,2)=1/6(sin^(3)xcos^(3)x-3I_(2,2))` `I_(4,2)=1/6(sin^(3)xcos^(3)x-3I_(2,2))` SOLUTION :A

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