1. **State the problem:** Evaluate the integral $$\int \frac{dx}{x^2 + 12x + 45}$$. 2. **Rewrite the quadratic expression:** Complete the square for the denominator: $$x^2 + 12x + 45 = (x^2 + 12x + 36) + 9 = (x + 6)^2 + 3^2$$. 3. **Recall the integral formula:** For constants $a$ and $b$, $$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$$. 4. **Apply substitution:** Let $u = x + 6$, so $du = dx$. The integral becomes $$\int \frac{du}{u^2 + 3^2}$$. 5. **Use the formula:** Substitute $a = 3$: $$\int \frac{du}{u^2 + 3^2} = \frac{1}{3} \arctan\left(\frac{u}{3}\right) + C$$. 6. **Back-substitute:** Replace $u$ with $x + 6$: $$\frac{1}{3} \arctan\left(\frac{x + 6}{3}\right) + C$$. **Final answer:** $$\int \frac{dx}{x^2 + 12x + 45} = \frac{1}{3} \arctan\left(\frac{x + 6}{3}\right) + C$$. This matches the bottom-right box in the image, confirming the correct integral evaluation.