The value of $\underset{\sqrt{\log 2}}{\overset{\sqrt{\log 3}}{\int }}\frac{x\sin x^2}{\sin x^2+\sin \left(\log 6-x^2\right)}dx$ is

$I=\underset{\sqrt{\log 2}}{\overset{\sqrt{\log 3}}{\int }}\frac{x\sin x^2}{Sm{x}^2+1Sin\left(\log 6-x^2\right)}dx$

$x^2=t$                          $dx=\frac{1}{2}dt$

$=\frac{1}{2}\underset{\log 2}{\overset{\log 3}{\int }}\frac{S\mathrm{int}}{S\mathrm{in}-1\sin \left(\log 6-t\right)}dt$

$\underset{a}{\overset{b}{\int }}f\left(x\right)\lambda =\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)$

$2I=\frac{1}{2}{\left(x\right)}_{\log 2}^{\log 3}$

$I=\frac{1}{4}\log \frac{3}{2}$