1. **State the problem:** We need to evaluate the integral $$\int \frac{2}{9 + x^2} \, dx$$. 2. **Recall the formula:** The integral of $$\frac{1}{a^2 + x^2}$$ with respect to $$x$$ is $$\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$$, where $$a > 0$$. 3. **Identify constants:** Here, $$a^2 = 9$$ so $$a = 3$$. 4. **Rewrite the integral:** $$\int \frac{2}{9 + x^2} \, dx = 2 \int \frac{1}{9 + x^2} \, dx$$ 5. **Apply the formula:** $$2 \int \frac{1}{9 + x^2} \, dx = 2 \cdot \frac{1}{3} \arctan\left(\frac{x}{3}\right) + C$$ 6. **Simplify the coefficient:** $$2 \cdot \frac{1}{3} = \frac{2}{3}$$ 7. **Final answer:** $$\int \frac{2}{9 + x^2} \, dx = \frac{2}{3} \arctan\left(\frac{x}{3}\right) + C$$ This means the integral evaluates to $$\frac{2}{3}$$ times the arctangent of $$\frac{x}{3}$$ plus the constant of integration.