1. **State the problem:** Evaluate the definite integral $$\int_{-\infty}^{\infty} \frac{x}{(x^2 + 9)^{3/2}} \, dx$$. 2. **Recall the properties:** The integrand is an odd function because the numerator $x$ is odd and the denominator $(x^2 + 9)^{3/2}$ is even. Specifically, $$f(-x) = \frac{-x}{((-x)^2 + 9)^{3/2}} = -\frac{x}{(x^2 + 9)^{3/2}} = -f(x).$$ 3. **Important rule:** The integral of an odd function over symmetric limits $[-a, a]$ is zero, i.e., $$\int_{-a}^a f(x) \, dx = 0$$ if $f$ is odd. 4. **Apply the rule:** Since the limits are from $-\infty$ to $\infty$ and the function is odd, the integral evaluates to zero without further calculation. 5. **Final answer:** $$\int_{-\infty}^{\infty} \frac{x}{(x^2 + 9)^{3/2}} \, dx = 0.$$