Determine if the improper integral is convergent or divergent, and calculate its value if it is convergent. \[ \int_{0}^{\infty} e^{6 x} \mathrm{dx} \] Calculate the value of the improper integral. Select the correct choice below and fill in any answer boxes within your choice. A. \[ \int_{0}^{\infty} e^{6 x} \mathrm{dx}=\square \text { (Type an integer or a fraction. Simplify your answer.) } \] B. The integral diverges.

Step 1 ：The integral is an improper integral from 0 to infinity of the function \(e^{6x}\). To determine if it is convergent or divergent, we need to calculate the integral and see if it results in a finite number (convergent) or not (divergent).

Step 2 ：The antiderivative of \(e^{6x}\) is \(\frac{1}{6}e^{6x}\). We need to evaluate this from 0 to infinity.

Step 3 ：If the limit as x approaches infinity of \(\frac{1}{6}e^{6x} - \frac{1}{6}e^{0}\) is finite, then the integral is convergent. If not, it is divergent.

Step 4 ：The limit as x approaches infinity of \(\frac{1}{6}e^{6x}\) is infinity, which means the integral is divergent.

Step 5 ：\(\boxed{\text{The integral diverges.}}\)