Evaluate the following integral or state that it diverges. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The improper integral converges and \( \frac{d x}{x^{2}+361} \frac{d x}{x^{2}+361}=\square \). (Type an exact answer, B. The improper integral diverges.

To evaluate the improper integral \(\int \frac{dx}{x^2 + 361}\), we can use the fact that \(x^2 + 361 = x^2 + 19^2\). The integral can then be solved using the formula for the integral of a secant function: \[ \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C. \] In this case, \(a = 19\). Therefore, we have: \[ \int \frac{dx}{x^2 + 361} = \frac{1}{19} \tan^{-1}\left(\frac{x}{19}\right) + C. \] Since the limit of \(\tan^{-1}\) approaches \( \frac{\pi}{2} \) as \(x\) approaches infinity, the integral converges. Thus, the answer would be: A. The improper integral converges and \( \frac{d x}{x^{2}+361} = \frac{1}{19} \tan^{-1}\left(\frac{x}{19}\right) + C \).