Evaluate $\int_{\sqrt{\ln2}}^{\sqrt{\ln3}}\frac{x\sin\left ( x^{2} \right )}{\sin\left ( x^{2} \right )+\sin\left ( \ln6-x^{2} \right )}dx$ How to evaluate
$$I=\int_{\sqrt{\ln2}}^{\sqrt{\ln3}}\frac{x\sin\left ( x^{2} \right )}{\sin\left ( x^{2} \right )+\sin\left ( \ln6-x^{2} \right )}\,\mathrm dx$$
I tried to substitute $x^2=t$ so
$$I=\frac{1}{2}\int_{\ln2}^{\ln3}\frac{\sin t}{\sin t+\sin\left ( \ln6-t \right )}\, \mathrm{d}t
$$
but I got stuck.Any hint?Thx!

Use the fact $\displaystyle\int_{a}^{b}f\left ( x \right )\mathrm{d}x=\int_{a}^{b}f\left ( a+b-x \right )\mathrm{d}x$ here we get
$$I=\frac{1}{2}\int_{\ln2}^{\ln3}\frac{\sin \left ( \ln6-t \right )}{\sin t+\sin\left ( \ln6-t \right )}\, \mathrm{d}t$$
hence
$$2I=\frac{1}{2}\int_{\ln2}^{\ln3}\, \mathrm{d}t$$
and the answer will follow.
