JEE Math Practice Question

Let the solution curve $$y=f(x)$$ of the differential equation $$ \frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}$$, $$x\in(-1,1)$$ pass through the origin. Then $$\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) d x $$ is equal to

1. A.$$\frac{\pi}{3}-\frac{1}{4}$$
2. B.$$\frac{\pi}{3}-\frac{\sqrt{3}}{4}$$
3. C.$$\frac{\pi}{6}-\frac{\sqrt{3}}{4}$$
4. D.$$\frac{\pi}{6}-\frac{\sqrt{3}}{2}$$