1. **State the problem:** We need to evaluate the integral $$\int \frac{9x}{3x^{2}+k} \, dx$$ where $k$ is a constant. 2. **Identify the formula and method:** This is a rational function where the numerator is the derivative of the denominator up to a constant factor. Recall that $$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C$$. 3. **Check if numerator matches derivative of denominator:** The denominator is $3x^{2}+k$. Its derivative is $$\frac{d}{dx}(3x^{2}+k) = 6x$$. 4. **Rewrite the integral:** The numerator is $9x$, which can be written as $$9x = \frac{9}{6} \cdot 6x = \frac{3}{2} \cdot 6x$$. 5. **Substitute and integrate:** $$\int \frac{9x}{3x^{2}+k} \, dx = \int \frac{\frac{3}{2} \cdot 6x}{3x^{2}+k} \, dx = \frac{3}{2} \int \frac{6x}{3x^{2}+k} \, dx$$ Using the formula, this becomes: $$\frac{3}{2} \ln|3x^{2}+k| + C$$ 6. **Final answer:** $$\boxed{\frac{3}{2} \ln|3x^{2}+k| + C}$$