1. The problem is to evaluate the integral $$\int \frac{9}{2} \sqrt{x} \ln x \, dx$$ and match it with one of the given options. 2. Rewrite the integral in a simpler form: $$\int \frac{9}{2} x^{1/2} \ln x \, dx$$. 3. Use integration by parts formula: $$\int u \, dv = uv - \int v \, du$$. 4. Choose $$u = \ln x$$ and $$dv = \frac{9}{2} x^{1/2} dx$$. 5. Then, $$du = \frac{1}{x} dx$$ and $$v = \frac{9}{2} \int x^{1/2} dx = \frac{9}{2} \cdot \frac{2}{3} x^{3/2} = 3 x^{3/2}$$. 6. Apply integration by parts: $$\int \frac{9}{2} x^{1/2} \ln x \, dx = 3 x^{3/2} \ln x - \int 3 x^{3/2} \cdot \frac{1}{x} dx = 3 x^{3/2} \ln x - 3 \int x^{1/2} dx$$. 7. Evaluate the remaining integral: $$\int x^{1/2} dx = \frac{2}{3} x^{3/2}$$. 8. Substitute back: $$3 x^{3/2} \ln x - 3 \cdot \frac{2}{3} x^{3/2} + C = 3 x^{3/2} \ln x - 2 x^{3/2} + C$$. 9. Factor out $$x^{3/2}$$: $$x^{3/2} (3 \ln x - 2) + C$$. 10. Note that $$3 \ln x = \ln x^3$$, so the expression can be written as: $$x^{3/2} (\ln x^3 - 2) + C$$. 11. Comparing with the options, the correct answer is option (a): $$x^{3/2} (\ln x^3 - 2) + C$$.