For `x in R and a` continuous function f(x) , let `I_(1)int_(sin^(2)t)^(1+cos^(2)t) xf{x(2-x)} dx and I_(2) I_(1)int_(sin^(2)t)^(1+cos^(2)t) f(x(2-x))dx.` Then, `(I_(1))/(I_(2))`=

Options

A. 0
B. 1
C. 2
D. 3

Answer

B

Explanation

We have,
`I_(1)underset(sin^(2)t)overset(1+cos^(2)t)int f(x)dx=underset(a)overset(b)intf(a+b-x)dx`
`:.I_(1)=underset(sin^(2)t)overset(1+cos^(2)t)int xf{x(2-x)}dx`
`rArr I_(1)=underset(sin^(2)t)overset(1+cos^(2)t)int (2-x)f{x(2-x)}dx`
`rArr I_(1)=2underset(sin^(2)t)overset(1+cos^(2)t)int f{x(2-)}dx-underset(sin^(2)t)overset(1+cos^(2)t)int xf{x(2-)}dx`
`rArr I_(1)=2I_(2)-I_(1)`
`rArr 1I_(1)=2I_(2)rArr (I_(1))/(I_(2))=1`