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For x epsilonR, and a continuous functio...

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

A

0

B

1

C

2

D

3

Text Solution

Verified by Experts

The correct Answer is:
B
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`
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