Problem 48 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(41-60\) $$\int \operatorname{csch}^{2}(5-x) d x$$

Step-by-Step Solution

Answer

The integral evaluates to \( \operatorname{coth}(5-x) + C \).

Step 1: Identify the Integral Type

The given integral \( \int \operatorname{csch}^{2}(5-x) \, dx \) involves the hyperbolic cosecant function squared. Generally, with hyperbolic functions, looking for derivatives of the function can simplify finding integrals.

Step 2: Recall the Derivative of Hyperbolic Cotangent

Recall the identity \( \frac{d}{dx} \operatorname{coth}(x) = -\operatorname{csch}^2(x) \). This means that integrating \( \operatorname{csch}^2(x) \) will yield \( -\operatorname{coth}(x) + C \) where \( C \) is the constant of integration.

Step 3: Apply Substitution for the Integration

To integrate \( \operatorname{csch}^2(5-x) \), we perform a substitution. Let \( u = 5 - x \), then \( du = -dx \). The integral becomes:\[ \int \operatorname{csch}^2(u)(-du) = -\int \operatorname{csch}^2(u) \, du. \]

Step 4: Integrate Using Known Identity

Using the derivative identity from Step 2, the integral of \(-\operatorname{csch}^2(u)\) is:\[ \int -\operatorname{csch}^2(u) \, du = \operatorname{coth}(u) + C. \]

Step 5: Substitute Back to Original Variable

Substitute \( u = 5 - x \) back into the result from Step 4:\[ \operatorname{coth}(u) + C = \operatorname{coth}(5-x) + C. \]

Step 6: Write the Final Answer

Thus, the final answer for the original integral is:\[ \int \operatorname{csch}^2(5-x) \, dx = \operatorname{coth}(5-x) + C. \]

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions, but they are defined using exponential functions instead. Just as trigonometric functions are related to circles, hyperbolic functions relate to hyperbolas in the Cartesian plane.
Some common hyperbolic functions include:

• Hyperbolic sine: \ \( \sinh(x) = \frac{e^x - e^{-x}}{2} \ \)
• Hyperbolic cosine: \ \( \cosh(x) = \frac{e^x + e^{-x}}{2} \ \)
• Hyperbolic tangent: \ \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \ \)
• Hyperbolic secant: \ \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \ \)
• Hyperbolic cosecant: \ \( \operatorname{csch}(x) = \frac{1}{\sinh(x)} \ \)
• Hyperbolic cotangent: \ \( \operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)} \ \)

Hyperbolic cosecant, which appears in the exercise, is the reciprocal of hyperbolic sine. These functions are essential in various applications including engineering, physics, and calculus because they provide solutions that model certain types of physical phenomena.

Substitution Method

Substitution is a fundamental technique in calculus, particularly useful for simplifying the process of finding integrals. The main idea is to change variables to make the integral easier to solve.
In the case of the example problem, the substitution method was used by letting \ \( u = 5 - x \ \), which simplifies the integral of \ \( \operatorname{csch}^2(5-x) \ \). Substitution also requires you to replace \ \( dx \ \) with \ \( du \ \) using the derivative of the substituted variable:

• Differential identity: \ \( du = -dx \ \)

This method is based on the chain rule for derivatives and allows for a straightforward integration process once the variable is changed. It's a vital tool for students because it helps to reduce complex integrals into forms that are easier to evaluate.

Cosecant Function

In mathematics, the cosecant function is one of the basic trigonometric functions. While it is more common in the realm of circles, it also has a hyperbolic counterpart used in integral calculus.
The hyperbolic cosecant, \ \( \operatorname{csch}(x) \ \), is defined as the reciprocal of the hyperbolic sine function:

• Formula: \ \( \operatorname{csch}(x) = \frac{1}{\sinh(x)} \ \)

Understanding the hyperbolic cosecant function is crucial when dealing with integrals involving reciprocal hyperbolic functions. Moreover, appreciating how it relates to its derivative, such as in the identity \ \( \frac{d}{dx} \operatorname{coth}(x) = -\operatorname{csch}^2(x) \ \), forms the foundation for effectively solving related integrals in calculus.

Integration Techniques

Integration techniques encompass a variety of methods used to evaluate integrals, which are vital operations in calculus. These techniques can simplify complex integrals substantially.
Some common integration techniques include:

• Substitution, as used in the problem above, which simplifies integrals by changing variables.
• Integration by parts, where integrals are broken down into simpler parts, often used for products of functions.
• Partial fraction decomposition, particularly for rational functions, which rewrites a function as a sum of simpler fractions.
• Trigonometric identities and integration, beneficial for integrating trigonometric functions, including hyperbolic ones.

Integration is the reverse process of differentiation. In this solution, recognizing that the derivative of the hyperbolic cotangent function is related to the hyperbolic cosecant squared is key. Mastering these techniques expands a student's problem-solving toolbox, making many seemingly complicated integrals accessible.