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

Verified step by step guidance

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

Key Concepts

Improper Integrals

Improper integrals involve integration over infinite intervals or integrands with infinite discontinuities. To evaluate them, limits are used to approach the problematic points, determining if the integral converges (has a finite value) or diverges (does not).

Convergence and Divergence of Integrals

An integral converges if its limit exists and is finite; otherwise, it diverges. For integrals over infinite intervals, convergence depends on the behavior of the integrand as the variable approaches infinity or negative infinity.

Behavior of Exponential Functions

Exponential functions like e^x and e^(-x) grow or decay rapidly. Understanding their limits as x approaches ±∞ helps analyze the integrand's behavior, which is crucial for determining the convergence of the integral.