Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))

Accepted Answer

Rewrite the integral to better understand the integrand. Notice that $$e^x + e$$^{-x}$$ can be expressed in terms of hyperbolic cosine: e^x$$ + $$e^{-x} = 2$$\cosh(x)$$. So the integral becomes $$\int_{-\infty}^{\infty} \frac{dx}{2\cosh(x)}$$. Recognize that the integrand $$\frac{1}{2\cosh(x)}$$ is an even function because $$\cosh(x)$$ is even. This allows us to simplify the integral over $$(-\infty, \infty)$$ to twice the integral over (0$$, \infty)$$: 2$$ \$$int_0$$^{\infty} \frac{dx}{2\cosh(x)} = \$$int_0$$^{\infty} \frac{dx}{\cosh(x)}$$. To test for convergence, analyze the behavior of the integrand as x$$ 	o \infty$$. Since $$\cosh(x) \approx \frac{e^x}{2}$$ for large x$, the integrand behaves like $$\frac{1}{\cosh(x)} \approx $$2e^{-x}$$, which is similar to an exponential decay function. Use the Direct Comparison Test by comparing $$\frac{1}{\cosh(x)} to$$ $$2e^{-x}$$ for large $$x. $$Since $$\$$int_0$$^{\infty} e$$^{-x}$$ dx$$ converges, and $$\frac{1}{\cosh(x)} \leq 2e$$^{-x}$$ for sufficiently large x$, the integral converges on (0$$, \infty)$$. Similarly, analyze the behavior as x$$ 	o 0$$ and x$$ 	o -\infty$$. Near zero, $$\cosh(x)$$ is finite and positive, so no issues with convergence there. As x$$ 	o -\infty$$, use the evenness of the function or the same comparison to 2e$$^{x}$$ (since $$\cosh(x) = \cosh(-x)$$). This confirms convergence over the entire real line.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))

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In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))

Risposta video verificata per un problema simile:

Concetti chiave

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))