Problem 51 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-60\) $$\int_{\ln 2}^{\ln 4} \operatorname{coth} x d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln 2.5 \).

Step 1: Understand the Integrand

The integrand given is \( \operatorname{coth} x \), which is the hyperbolic cotangent function, defined as \( \operatorname{coth} x = \frac{\cosh x}{\sinh x} \). This function can be tricky to integrate directly.

Step 2: Find an Appropriate Method

To integrate \( \operatorname{coth} x \), recognize that its derivative might suggest a substitution or direct integration. We utilize the fact that the integral of \( \operatorname{coth} x \) results in a natural logarithm, specifically, \( \int \operatorname{coth} x \, dx = \ln|\sinh x| + C \).

Step 3: Apply the Integral Formula

Now, apply the integration rule to the limits given: \( \int_{\ln 2}^{\ln 4} \operatorname{coth} x \, dx = \left[ \ln|\sinh x| \right]_{\ln 2}^{\ln 4} \).

Step 4: Evaluate at Upper Limit

First, calculate \( \ln|\sinh(\ln 4)| \). Utilize the identity \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), hence: \( \sinh(\ln 4) = \frac{4 - \frac{1}{4}}{2} = \frac{15}{8} \), thus: \( \ln|\sinh(\ln 4)| = \ln\left(\frac{15}{8}\right) \).

Step 5: Evaluate at Lower Limit

Now calculate \( \ln|\sinh(\ln 2)| \). Using the same identity: \( \sinh(\ln 2) = \frac{2 - \frac{1}{2}}{2} = \frac{3}{4} \), thus: \( \ln|\sinh(\ln 2)| = \ln\left(\frac{3}{4}\right) \).

Step 6: Compute Definite Integral Result

Subtract the value at the lower limit from that at the upper limit: \( \ln\left(\frac{15}{8}\right) - \ln\left(\frac{3}{4}\right) \). This simplifies to: \( \ln\left(\frac{15}{8} \times \frac{4}{3}\right) = \ln 2.5 \).

Step 7: Final Step: Conclusion

The definite integral of \( \int_{\ln 2}^{\ln 4} \operatorname{coth} x \, dx \) evaluates to \( \ln 2.5 \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas rather than circles. One of the key hyperbolic functions is the hyperbolic cotangent, denoted as \( \operatorname{coth} x \). It is defined by the formula:

• \( \operatorname{coth} x = \frac{\cosh x}{\sinh x} \)

where \( \cosh x \) and \( \sinh x \) are the hyperbolic cosine and sine, respectively. Unlike their trigonometric counterparts, hyperbolic functions have different properties, especially concerning their asymptotic behavior. The function \( \operatorname{coth} x \) is particularly useful in calculus when dealing with certain integrals.
It is important to be familiar with the identities and derivatives of these functions, as they can provide a simpler approach for integration or other calculus operations. Recognizing the role of \( \operatorname{coth} x \) can simplify solving problems involving hyperbolic functions.

Natural Logarithm

The natural logarithm, represented as \( \ln x \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is a fundamental function in calculus and appears frequently in solutions involving integration and differentiation.
For the exercise at hand, the natural logarithm emerges as a part of the integral of the hyperbolic cotangent function. Specifically, the integral of \( \operatorname{coth} x \) results in \( \ln|\sinh x| \).

• The relation between the hyperbolic functions and the natural logarithm is crucial to solve the definite integral.

Understanding the properties and rules of logarithms, such as the product, quotient, and power rules, aids in simplifying and computing expressions effectively. Hence, the natural logarithm simplifies the expression within the integral's evaluation.

Integration Techniques

Integration techniques are essential for solving a wide variety of calculus problems. In this specific task, integrating the hyperbolic cotangent \( \operatorname{coth} x \) requires recognizing that its integral simplifies to a natural logarithm function.
Some general integration techniques include:

• Substitution: Useful when the integrand is a composite function.
• Integration by Parts: Handy for products of functions.
• Recognizing Derivatives: Utilizing known integrals and derivatives can save time.

For \( \operatorname{coth} x \), knowing its integral directly is a significant shortcut. This knowledge reduces the need for more complex techniques, streamlining the process and leading directly to a natural logarithm expression. Mastery of such techniques improves problem-solving efficiency and aids in tackling more challenging integrals.

Definite Integrals Evaluation

Definite integrals are used to find the net area under a curve between specified limits. To evaluate a definite integral, one must compute the integral of the function and then apply the limits of integration.
In our specific exercise, we initially integrated \( \operatorname{coth} x \) and obtained \( \ln|\sinh x| \). The next step is to evaluate this expression at the given bounds \( \ln 2 \) and \( \ln 4 \).

• First, calculate \( \ln|\sinh(\ln 4)| \).
• Then compute \( \ln|\sinh(\ln 2)| \).
• Finally, subtract the latter from the former.

These calculations yield the expression \( \ln \left( \frac{15/8}{3/4} \right) \) when simplified. Recognizing how definite integrals convert these limits to a single value, in this case, \( \ln 2.5 \), underscores their usage in solving area and value problems within calculus efficiently.