Question:

<p><strong>Step 1:</strong> Differentiate both sides with respect to x using Leibniz rule (chain rule for integrals):</p><p>$$\frac{d}{dx}\int_0^{\sin x} f(t)\,dt = f(\sin x) \cdot \cos x$$</p><p><strong>Step 2:</strong> Differentiate the right side:</p><p>$$\frac{d}{dx}\left(\frac{\sqrt{3}}{2}x\right) = \frac{\sqrt{3}}{2}$$</p><p><strong>Step 3:</strong> Equate the derivatives:</p><p>$$f(\sin x) \cdot \cos x = \frac{\sqrt{3}}{2}$$</p><p><strong>Step 4:</strong> To find $f\left(\frac{\sqrt{3}}{2}\right)$, we need $\sin x = \frac{\sqrt{3}}{2}$, which occurs at $x = \frac{\pi}{3}$.</p><p><strong>Step 5:</strong> At $x = \frac{\pi}{3}$: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$</p><p><strong>Step 6:</strong> Substitute into the equation:</p><p>$$f\left(\frac{\sqrt{3}}{2}\right) \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}$$</p><p>$$f\left(\frac{\sqrt{3}}{2}\right) = \sqrt{3}$$</p><p>∴ Answer: <strong>B</strong></p>