1. **State the problem:** We need to evaluate the integral $$\int \frac{x}{4x^2+9} \, dx$$. 2. **Identify the formula and substitution:** Notice the denominator is a quadratic expression. We can use substitution to simplify the integral. Let $$u = 4x^2 + 9$$. 3. **Calculate the differential:** Differentiating, $$du = 8x \, dx$$, so $$x \, dx = \frac{du}{8}$$. 4. **Rewrite the integral:** Substitute into the integral: $$\int \frac{x}{4x^2+9} \, dx = \int \frac{1}{u} \cdot \frac{du}{8} = \frac{1}{8} \int \frac{1}{u} \, du$$. 5. **Integrate:** The integral of $$\frac{1}{u}$$ is $$\ln|u|$$, so $$\frac{1}{8} \int \frac{1}{u} \, du = \frac{1}{8} \ln|u| + C$$. 6. **Back-substitute:** Replace $$u$$ with $$4x^2 + 9$$: $$\frac{1}{8} \ln|4x^2 + 9| + C$$. **Final answer:** $$\int \frac{x}{4x^2+9} \, dx = \frac{1}{8} \ln|4x^2 + 9| + C$$