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Q.
The value of $\int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^{2}}{\sin x^{2}+\sin \left(\ln 6-x^{2}\right)} d x$ is
Integrals
Solution:
$
I=\frac{1}{2} \int_{\sqrt{\ln 2}}^{\sqrt{\ell n 3}} \frac{2 x \sin x^{2}}{\sin x^{2}+\sin \left(\ln 6-x^{2}\right)} d x
$
Let $x^{2}=t \Rightarrow 2 x d x=d t$
Also, when $x=\sqrt{\ell n 2}, t=\ell n 2$
when $x=\sqrt{\ln 3}, t=\ell n 3$
$
\therefore I=\frac{1}{2} \int_{\ell n 2}^{\ell n 3} \frac{\sin t d t}{\sin t+\sin (\ell n 6-t)} \ldots(1)
$
$
I=\frac{1}{2} \int_{\ln 2}^{\ln 3} \frac{\sin (\ell n 6-t)}{\sin t+\sin (\ln 6-t)} d t \ldots(2)
$
Adding values of $I$ in equations (1) and (2)
$
2 I=\frac{1}{2} \int_{\ell n 2}^{\ln 3} 1 d t=\frac{1}{2}(\ln 3-\ln 2)=\frac{1}{2} \ell n \frac{3}{2} \Rightarrow I=\frac{1}{4} \ell n \frac{3}{2}
$