Let the solution curve \( y=f(x) \) of the differential equation \( \frac{d y}{d x}+\frac{x y}{x...

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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 \( \int_{\frac{-\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) d x \) is
(a) \( \frac{\pi}{3}-\frac{1}{4} \)
(b) \( \frac{\pi}{3}-\frac{\sqrt{3}}{4} \)
(c) \( \frac{\pi}{6}-\frac{\sqrt{3}}{4} \)
(d) \( \frac{\pi}{6}-\frac{\sqrt{3}}{2} \)
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