Evaluate the following integral or state that it diverges. \( \int_{-\infty}^{\infty} \frac{d x}{x^{2}+361} \)

To evaluate the integral \( \int_{-\infty}^{\infty} \frac{dx}{x^{2}+361} \), we can use a known formula for integrals of this type. The integral of the form \( \int_{-\infty}^{\infty} \frac{dx}{x^{2} + a^{2}} \) is equal to \( \frac{\pi}{a} \) where \( a > 0 \). Here, \( a^{2} = 361 \) implies \( a = \sqrt{361} = 19 \). Applying the formula: \[ \int_{-\infty}^{\infty} \frac{dx}{x^{2}+361} = \frac{\pi}{19} \] Thus, the integral converges, and we find that: \[ \int_{-\infty}^{\infty} \frac{dx}{x^{2}+361} = \frac{\pi}{19} \]