13. Multiple Choice 1 point Determine if the integral diverges or converges. Evaluate the integral if it converges. \( \int_{0}^{\infty} \frac{1}{e^{+}} d x \) \( \pi / 2 \) \( \pi / 4 \) This integral diverges. \( \pi \)

To determine whether the integral \[ \int_{0}^{\infty} \frac{1}{e^{x}} \, dx \] converges or diverges, we can evaluate it step by step. 1. **Set up the integral**: We will evaluate the improper integral by taking the limit as \( b \) approaches infinity: \[ \int_{0}^{\infty} \frac{1}{e^{x}} \, dx = \lim_{b \to \infty} \int_{0}^{b} \frac{1}{e^{x}} \, dx \] 2. **Evaluate the integral**: The integral of \( \frac{1}{e^{x}} \) can be rewritten as \( e^{-x} \). The antiderivative of \( e^{-x} \) is \( -e^{-x} \). Thus, we have: \[ \int \frac{1}{e^{x}} \, dx = -e^{-x} + C \] 3. **Apply the limits**: Now we evaluate the definite integral from \( 0 \) to \( b \): \[ \int_{0}^{b} \frac{1}{e^{x}} \, dx = \left[-e^{-x}\right]_{0}^{b} = -e^{-b} - (-e^{0}) = -e^{-b} + 1 \] 4. **Take the limit**: Now we take the limit as \( b \) approaches infinity: \[ \lim_{b \to \infty} (-e^{-b} + 1) = \lim_{b \to \infty} (-0 + 1) = 1 \] Since the limit exists and is finite, the integral converges. Thus, the value of the integral is: \[ \int_{0}^{\infty} \frac{1}{e^{x}} \, dx = 1 \] Now, let's check the multiple-choice options provided: - \( \pi / 2 \) - \( \pi / 4 \) - This integral diverges. - \( \pi \) None of these options match the result of the integral, which is \( 1 \). Therefore, the correct conclusion is that the integral converges, but the provided options do not include the correct answer.