The value of I=∫ln⁡2ln⁡3 xsin⁡x2sin⁡x2+sin⁡ln⁡6−x2dx is

Let $I = \int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^{2}}{\sin x^{2} + \sin (\ln 6 - x^{2})} d x$

Putting $x^{2} = t$ , we get

$I = \frac{1}{2} \int_{\ln 2}^{\ln 3} \frac{\sin t}{\sin t + \sin (\ln 6 - t)} d t$ …(i)

Using $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x,$ we get

$I = \frac{1}{2} \int_{\ln 2}^{\ln 3} \frac{\sin (\ln 6 - t)}{\sin (\ln 6 - t) + \sin t} d t$ …(ii)

Adding (i) and (ii), we get

$2 I = \frac{1}{2} (\ln 3 - \log 2) = \frac{1}{2} \ln (\frac{3}{2}) \Rightarrow I = \frac{1}{4} \ln (\frac{3}{2})$

The value of $I = \int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^{2}}{\sin x^{2} + \sin (\ln 6 - x^{2})} d x$ is