1. The problem is to solve the first-order linear differential equation $$\frac{dy}{dx} + 3xy = \sin x.$$\n\n2. We are given the integrating factor $$u = e^{\frac{3}{2} x^{2}}.$$\n\n3. Multiply both sides of the differential equation by the integrating factor to get:\n$$e^{\frac{3}{2} x^{2}} \frac{dy}{dx} + 3x e^{\frac{3}{2} x^{2}} y = e^{\frac{3}{2} x^{2}} \sin x.$$\n\n4. Notice the left side is the derivative of $$y e^{\frac{3}{2} x^{2}}$$ with respect to $$x$$:\n$$\frac{d}{dx} \left(y e^{\frac{3}{2} x^{2}}\right) = e^{\frac{3}{2} x^{2}} \sin x.$$\n\n5. Integrate both sides with respect to $$x$$:\n$$y e^{\frac{3}{2} x^{2}} = \int e^{\frac{3}{2} x^{2}} \sin x \, dx + C,$$\nwhere $$C$$ is the constant of integration.\n\n6. The integral $$\int e^{\frac{3}{2} x^{2}} \sin x \, dx$$ does not have a simple closed form in elementary functions, so the solution is expressed implicitly as:\n$$y = e^{-\frac{3}{2} x^{2}} \left( \int e^{\frac{3}{2} x^{2}} \sin x \, dx + C \right).$$\n\n7. Summary: The solution to the differential equation is\n$$\boxed{y = e^{-\frac{3}{2} x^{2}} \left( \int e^{\frac{3}{2} x^{2}} \sin x \, dx + C \right)}.$$