Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \operatorname{csch} x d x\)

Step-by-Step Solution

Answer

The integral converges to \(-\ln |\tanh(1/2)|\).

Step 1: Identify the Improper Integral

The given integral is \( \int_{1}^{\infty} \operatorname{csch} x \, dx \). Since the upper limit of integration is infinity, this is an improper integral. To evaluate it, we first rewrite it as a limit: \( \lim_{b \to \infty} \int_{1}^{b} \operatorname{csch} x \, dx \).

Step 2: Find the Antiderivative of \( \operatorname{csch} x \)

The function \( \operatorname{csch} x = \frac{1}{\sinh x} \). The function \( \sinh x \) is an odd function related to hyperbolic sine, with the property that its derivative is \( \cosh x \). However, directly integrating \( \operatorname{csch} x \) involves using substitution or recognizing it in terms of known integrals, where \( \int \operatorname{csch} x \ dx = \ln |\tanh(x/2)| + C \).

Step 3: Evaluate the Definite Integral

Using the antiderivative from Step 2, evaluate the definite integral from 1 to \( b \): \( \int_{1}^{b} \operatorname{csch} x \, dx = \left[ \ln |\tanh(x/2)| \right]_{1}^{b} = \ln |\tanh(b/2)| - \ln |\tanh(1/2)| \).

Step 4: Take the Limit as \( b \to \infty \)

Take the limit of the expression \( \ln |\tanh(b/2)| - \ln |\tanh(1/2)| \) as \( b \to \infty \). As \( x \to \infty \), \( \tanh(x/2) \to 1 \), meaning \( \ln |\tanh(b/2)| \to \ln 1 = 0 \). Thus, the expression approaches \( 0 - \ln |\tanh(1/2)| \).

Step 5: Analyze Convergence

Since the integral evaluates to \( -\ln |\tanh(1/2)| \), which is a finite number, the improper integral \( \int_{1}^{\infty} \operatorname{csch} x \, dx \) converges.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles. One of the functions most commonly encountered in calculus is the hyperbolic sine function, denoted as \( \sinh(x) \). The hyperbolic cosecant function \( \operatorname{csch}(x) \) is defined as the reciprocal of \( \sinh(x) \):

• \( \operatorname{csch}(x) = \frac{1}{\sinh(x)} \)
• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)

These functions have properties similar to their trigonometric counterparts. For instance:

• \( \sinh(x) \) is an odd function, just like \( \sin(x) \).
• Its derivative, \( \cosh(x) \), is similar to \( \cos(x) \). It is worth noting that the integral of \( \operatorname{csch}(x) \) involves unique results such as \( \ln |\tanh(x/2)| + C \).

In practice, hyperbolic functions arise in various scenarios like modeling hyperbolic geometry, special relativity, and describing the shape of a hanging cable or chain, which form a catenary curve.

Convergence of Integrals

When we talk about convergence in the context of improper integrals, we refer to whether the integral has a finite value over an infinite interval. The integral \( \int_{1}^{\infty} \operatorname{csch}(x) \, dx \) is an improper integral because it extends to infinity.To evaluate an improper integral, we must perform a limit process. This often means converting the integral to a limit of finite integrals as some boundary approaches infinity. For example:

• \( \lim_{b \to \infty} \int_{1}^{b} \operatorname{csch} x \, dx \)

In this problem, we computed the antiderivative, evaluated at the upper bound, and took the limit as it approached infinity.The result \( -\ln |\tanh(1/2)| \) being a finite number indicates that this integral converges. If the result approached infinity or failed to exist, the integral would have diverged. The analysis of convergence is crucial in evaluating the viability of integrals in practical applications, ensuring they result in meaningful quantities.

Integration Techniques

Integration techniques encompass various methods to find antiderivatives or evaluate definite integrals. When dealing with functions like \( \operatorname{csch}(x) \), recognizing it as the reciprocal of \( \sinh(x) \) is the first step.Some techniques useful in integration include:

• Substitution: Useful when an integral contains a composite function. It can simplify the integral by substituting variables to reduce complexity.
• Integration by Parts: This technique is helpful when the integral is the product of functions and can be simplified by distributing differentiation and integration tasks across these functions.

For the integral \( \int \operatorname{csch}(x) \, dx \), an essential observation was using known integral forms such as \( \ln |\tanh(x/2)| + C \). Recognizing these forms can save time and effort in evaluation.Understanding when and how to apply specific techniques enables efficient and accurate calculation of integrals, which is fundamental in various fields, from physics to engineering.