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

To determine the behavior of the integral, we can simplify the integrand. Notice that the expression \( e^{x} + e^{-x} \) can be rewritten as \( 2 \cosh(x) \). Thus, we have: \[ \int_{0}^{\infty} \frac{1}{e^{x}+e^{-x}} \, dx = \int_{0}^{\infty} \frac{1}{2 \cosh(x)} \, dx. \] The integral converges, and evaluating it using the known result yields: \[ \int_{0}^{\infty} \frac{1}{\cosh(x)} \, dx = \frac{\pi}{2}, \] hence, the value of the original integral is: \[ \int_{0}^{\infty} \frac{1}{e^{x}+e^{-x}} \, dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4}. \] So, the answer is \( \pi / 4 \). Additionally, thinking about the convergence of this integral, you can explore various methods, such as the Comparison Test. Comparing with a simpler function that behaves similarly at infinity can help confirm convergence. On the evaluation side, if you ever get stuck integrating, consider using substitution methods or recognizing forms of integrals that yield known results, like those involving hyperbolic functions. They often pop up in various applications, making them quite handy in your mathematical toolbox!