The valued of `int_(sqrt(In2))^(sqrt(In3)) (x sinx^(2))/(sinx^(2)+sin(In 6-x^(2)))dx` is

Options

A. `(1)/(4)"In"(3)/(2)`
B. `(1)/(2)"In"(3)/(2)`
C. `"In"(3)/(2)`
D. `(1)/(6)"In"(3)/(2)`

Answer

A

Explanation

Let `I=overset(sqrt("In"3))underset(sqrt("In"2))int(xsinx^(2))/(sinx^(2)+sin(In6-x^(2)))dx`
Putting `x^(2)=t`, we get
`I=(1)/(2)overset("In"3)underset("In"2)int (sint)/(sint+sin(In6-t))dt" "`..........(i)
Using `overset(b)underset(a)int f(x)dx=overset(b)underset(a)int f(a+b-x)dx`, we get
`I=(1)/(2)overset("In"3)underset("In"2)int (Sin("In"6-t))/(sin("In"6-t)+sint)dt " "`......(ii)
Adding (i) and (ii), we get
`2I=(1)/(2)("In"3-log2)=(1)/(2)"In"((3)/(2))rArr I=(1)/(4)"In"((3)/(2))`