Problem 402 - Calculus Volume 1

Question

For the following exercises, find the antiderivatives for the given functions. $$\operatorname{coth}(x)$$

Step-by-Step Solution

Answer

Antiderivative is \( \ln|\sinh(x)| + C \).

Step 1: Understand the Function

The function given is the hyperbolic cotangent, written as \( \operatorname{coth}(x) \). It is defined as \( \operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)} \), where \( \cosh(x) \) and \( \sinh(x) \) are the hyperbolic cosine and sine functions, respectively.

Step 2: Recall the Relationship with Derivatives

Remember that the derivative of the logarithm of the absolute value of the hyperbolic sine function is \( \frac{1}{\sinh(x)} \), thus the derivative of \( \operatorname{coth}(x) \) forms when considering the integration of such expressions.

Step 3: Identify the Antiderivative Formula

The antiderivative of \( \operatorname{coth}(x) \) is related to the natural logarithm function. Specifically, the rule is that the antiderivative of \( \operatorname{coth}(x) \) is \( \ln|\sinh(x)| + C \), where \( C \) is the constant of integration.

Step 4: Verify the Antiderivative

Differentiate \( \ln|\sinh(x)| + C \) to check if it results in \(\operatorname{coth}(x)\). Differentiating \( \ln|\sinh(x)| \) using the chain and quotient rules gives \( \operatorname{coth}(x) \), confirming that the integration process is correct.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to the regular trigonometric functions but based on hyperbolas instead of circles.
They are crucial in various fields of mathematics and physics due to their unique properties. The two most basic hyperbolic functions are the hyperbolic sine, \(\sinh(x)\), and the hyperbolic cosine, \(\cosh(x)\).
These are defined as follows:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

The function \(\operatorname{coth}(x)\) is the hyperbolic cotangent and is given by the formula \(\operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\). This function bears similarities to its trigonometric counterpart, the regular cotangent function but applies to the hyperbolic sine and cosine instead. Understanding these functions is key to solving problems involving hyperbolic identities and integration.

Integration Techniques

Integration techniques are essential tools for finding antiderivatives, and several methods can be used depending on the structure of the function.
Some common techniques include substitution, integration by parts, and recognizing standard forms. In the case of \(\operatorname{coth}(x)\), recognizing its relation to the derivative of a logarithmic form is crucial.

• Substitution: Often used when a function is composed, allowing simplification by changing variables.
• Integration by Parts: Useful when integrating products of functions, based on the derivative and antiderivative relationship.
• Recognizing Standard Forms: Identifying known derivatives that are linked to familiar functions such as logarithms.

By understanding these techniques, one can efficiently approach integrals, especially those involving hyperbolic functions as in this case.

Logarithmic Integration

Logarithmic integration is a technique that plays a critical role in handling integrals involving hyperbolic functions. Specifically, \(\operatorname{coth}(x)\) is related to the natural logarithm.
This relationship is due to the identity \[ \int \operatorname{coth}(x)\, dx = \ln|\sinh(x)| + C \] where \(C\) is the constant of integration.
This integral reflects the property that the derivative of \(\ln|\sinh(x)|\) results in \(\operatorname{coth}(x)\).

• Logarithmic integration commonly arises when the integrand resembles the derivative of a logarithmic function.
• This technique benefits from understanding derivative rules and logarithmic properties.

Mastering logarithmic integration helps in solving complex integrals efficiently, especially when hyperbolic functions are involved.

Hyperbolic Cotangent

The hyperbolic cotangent, \(\operatorname{coth}(x)\), is a fundamental hyperbolic function defined as \(\operatorname{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\).
This function is analogous to the trigonometric cotangent and plays a similar role within hyperbolic identities and calculus.

• \(\operatorname{coth}(x)\) is used in the integration to find antiderivatives involving hyperbolic functions.
• The function is significant in situations involving differential equations, general relativity, and various physics applications.
• Understanding \(\operatorname{coth}(x)\) requires familiarity with both \(\sinh(x)\) and \(\cosh(x)\) since these are its building blocks.

The ability to work with \(\operatorname{coth}(x)\) and its properties is essential for advanced calculus and mathematical modeling.