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T h ev a l u eofint(sqrt(1n2))^(sqrt(1n3...

`T h ev a l u eofint_(sqrt(1n2))^(sqrt(1n3))(xsinx^2)/(sinx^2+sin(1n6-x^2))dxi s` `1/4 1n3/2` (b) `1/21n3/2` `1n3/2` (d) `1/61n3/2`

A

`(1)/(4)"log"(3)/(2)`

B

`(1)/(2)"log"(3)/(2)`

C

`"log"(3)/(2)`

D

`(1)/(6)"log"(3)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Put `x^(2)=trArrx dx=dt//2`
`:.I=int_(log2)^(log3)(sint*(dt)/(2))/(sint+sin(log6-t))` . . . (i)
Using , `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx`
`=(1)/(2)int_(log2)^(log3)(sin(log2+log3-t))/(sin(log2+log3-t)+sin(log6-(log2+log3-t)))`
`=(1)/(2)int_(log2)^(log3)(sin(log6-t))/(sin(log6-t)+sin(t))dt`
`:. I=int_(log2)^(log3)(sin(log6-t))/(sin(log6-t)+sint)dt` . . . (ii)
On adding Eqs . (i) and (ii) , we get
`2I=(1)/(2)int_(log2)^(log3)(sint+sin(log6-t))/(sin(log6-t)+sint)dt`
`rArr2I=(1)/(2)(t)_(log2)^(log3)=(1)/(2)(log3-log-2)`
`:.I=(1)/(4)log((3)/(2))`
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