$\text{ The function }\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)\text{ is the solution of the differential equation }$$\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{xy}}{{\mathrm{x}}^2-1}=\frac{{\mathrm{x}}^4+2\mathrm{x}}{\sqrt{1-{\mathrm{x}}^2}}\text{ in }\mathrm{x}\in (-1,1)\text{ satisfying }\mathrm{f}\left(0\right)=0\text{ the }{\int }_{-\frac{\sqrt{3}}{2}}^{\sqrt{3}/2} \mathrm{f}\left(\mathrm{x}\right)dx\text{ is }$

$\text{ I.F.= }{\mathrm{e}}^{\int \frac{\mathrm{x}}{{\mathrm{x}}^2-1}\mathrm{dx}}={\mathrm{e}}^{\frac{1}{2}\log \left({\mathrm{x}}^2-1\right)\mid }=\sqrt{1-{\mathrm{x}}^2}$

$\text{ Solution of D.E is }y\sqrt{1-x^2}=\int \frac{x^4+2x}{\sqrt{1-x^2}}\cdot \sqrt{1-x^2}dy=\frac{x^5}{5}+x^2+c$

$\begin{array}{l}\mathrm{f}\left(0\right)=0\Rightarrow \mathrm{c}=0\\ y\sqrt{1-{\mathrm{x}}^2}=\frac{{\mathrm{x}}^5}{5}+{\mathrm{x}}^2\\ {\int }_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} y \mathrm{dx}={\int }_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} \frac{{\mathrm{x}}^2}{\sqrt{1-{\mathrm{x}}^2}}\mathrm{dx} \\ If f(-x)=f\left(x\right) then {\int }_{-a}^{a}f\left(x\right) dx=2{\int }_0^{a}f\left(x\right) dx\\ =2{\int }_0^{\frac{\sqrt{3}}{2}} \frac{{\mathrm{x}}^2}{\sqrt{1-{\mathrm{x}}^2}}\mathrm{dx},\text{ put }\mathrm{x}=\sin \theta \Rightarrow dx=\cos \theta d\theta \\ =2{\int }_0^{\pi /3} {\sin }^2\theta \mathrm{d}\theta \\ =2\int \frac{1-\cos 2\theta }{2}d\theta \\ ={\left(\theta -\frac{\sin 2\theta }{2}\right)}_0^{\frac{\pi }{3}}\\ =\frac{\pi }{3}-\frac{\sqrt{3}}{4}\end{array}$