1. **Problem statement:** Calculate the integral $$\int_{-\infty}^{+\infty} \frac{dx}{x^2 + 2x + 2}$$ and study its convergence. 2. **Rewrite the integrand:** Complete the square in the denominator: $$x^2 + 2x + 2 = (x+1)^2 + 1$$ 3. **Recognize the integral form:** The integral becomes $$\int_{-\infty}^{+\infty} \frac{dx}{(x+1)^2 + 1}$$ 4. **Recall the standard integral formula:** For real $a > 0$, $$\int_{-\infty}^{+\infty} \frac{dx}{x^2 + a^2} = \frac{\pi}{a}$$ 5. **Apply the formula:** Here, $a = 1$, so $$\int_{-\infty}^{+\infty} \frac{dx}{(x+1)^2 + 1} = \pi$$ 6. **Convergence:** The integral converges because the integrand behaves like $1/x^2$ for large $|x|$, which is integrable over $\mathbb{R}$. **Final answer:** $$\int_{-\infty}^{+\infty} \frac{dx}{x^2 + 2x + 2} = \pi$$