Question

Which of the improper integrals in Exercises 63–68 converge and which diverge?∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx

Accepted Answer

First, recognize that the integral is improper because the limits of integration are from $$-\infty to$$ $$\infty. $$Rewrite the integrand $$\frac{2}{e^x + e$$^{-x}$$} in a $$simpler form. Notice that $$e^x + e$$^{-x}$$ = 2 \cosh x$$, so the integrand becomes $$\frac{2}{2 \cosh x} = \frac{1}{\cosh x}. $$Express the integral as $$\int_{-\infty}^{\infty} \frac{1}{\cosh x} \, dx. $$This is a standard integral involving the hyperbolic secant function, since $$\frac{1}{\cosh x} = \operatorname{sech} x. To $$determine convergence, analyze the behavior of $$\operatorname{sech} x as$$ $$x 	o \pm \infty. $$Since $$\cosh x$$ grows exponentially, $$\operatorname{sech} x$$ approaches zero rapidly, suggesting the integral may converge. Set up the integral as the sum of two limits: $$\lim_{a 	o -\infty} \int_a^0$ \operatorname{sech} x \, dx + \lim_{b 	o \infty} \$$int_0$$^b \operatorname{sech} x \, dx. $$Evaluate these integrals using the antiderivative of $$\operatorname{sech} x$$, which is $$2 \arctan(	anh(\frac{x}{2}))$$, to check for convergence.

Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx

Verified video answer for a similar problem:

Key Concepts

Improper Integrals

Convergence and Divergence of Integrals

Behavior of Exponential Functions