Question

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

Short answer

The antiderivative of \(\operatorname{coth}(x)\) is \(\ln|\sinh(x)| + C\).

Step-by-step solution

Understanding the Problem

We need to find the antiderivative, also known as the indefinite integral, of the function \(\operatorname{coth}(x)\). The function \(\operatorname{coth}(x)\) represents the hyperbolic cotangent function given by \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\).

Rewrite the Function

Given that \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\), we can rewrite it in terms of its basic hyperbolic identities. This helps simplify the integration process, as we know the differential of \(\ln|\sinh(x)|\) gives \(\coth(x)\).

Recognize the Integral Form

The integral of \(\coth(x)\) is a standard result from calculus: \[ \int \coth(x) \, dx = \ln|\sinh(x)| + C \] This comes from recognizing the differential form \(\frac{d}{dx}[\ln|\sinh(x)|] = \coth(x)\).

Write the Antiderivative

Using the standard result from calculus, we can directly write the antiderivative of \(\coth(x)\): \[ \int \coth(x) \, dx = \ln|\sinh(x)| + C \] Here, \(C\) is the constant of integration.

Key concepts

Indefinite Integral

In calculus, an indefinite integral is a fundamental concept that involves finding the antiderivative of a function, essentially reversing the process of differentiation. It is represented by the integral sign \(\int\), followed by the function and \(dx\), which indicates the variable of integration. Unlike definite integrals, indefinite integrals do not have specified limits; hence they include a constant of integration \(C\), representing an infinite family of functions.

Indefinite integrals are used to find functions whose derivative gives the original function inside the integral. For example, if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int f(x) \, dx = F(x) + C\). This concept is crucial in understanding how functions behave across various domains and in solving various mathematical and physical problems.

To find indefinite integrals, we often use several techniques such as substitution, integration by parts, and recognizing standard integral forms. In the context of the original exercise with \(\coth(x)\), recognizing that \(\int \coth(x)\, dx = \ln|\sinh(x)| + C\) is a key step in solving the integral for hyperbolic functions.

Hyperbolic Functions

Hyperbolic functions are analogs of the trigonometric functions but based on hyperbolas instead of circles. These functions include \(\sinh(x)\), \(\cosh(x)\), \(\tanh(x)\), \(\coth(x)\), and their inverses. They have properties similar to the trigonometric functions but are defined using exponential functions.

The hyperbolic cotangent function, \(\coth(x)\), is defined as \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\). This function is essential in calculus as it appears in various applications, such as solving differential equations and evaluating integrals involving hyperbolic identities.

Hyperbolic functions have unique properties:

• They can be expressed through exponential functions: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\) and \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).
• Their derivatives mirror trigonometric identities: \(\frac{d}{dx}[\sinh(x)] = \cosh(x)\) and \(\frac{d}{dx}[\cosh(x)] = \sinh(x)\).

Understanding these properties allows us to simplify problems involving hyperbolic functions and helps in forming connections with their trigonometric counterparts.

Calculus

Calculus is the branch of mathematics that studies change, and it plays a critical role in various scientific fields. It is divided into two main branches: differential calculus and integral calculus. Differential calculus concerns itself with derivatives and the rates of change, while integral calculus focuses on antiderivatives and areas under curves.

Integral calculus, in particular, involves finding integrals, which can be either definite or indefinite. Definite integrals, which have limits, calculate the accumulation of quantities, such as areas and total distance. Indefinite integrals, on the other hand, seek to determine functions that represent antiderivatives.

Using calculus, we can solve a broad range of problems:

• Determine the motion paths in physics by integrating velocity functions to find displacement.
• Find the accumulated growth in biological systems by solving differential equations.
• Understand economic models through integration to optimize functions describing supply and demand.

In solving the antiderivative for \(\coth(x)\) as in the problem provided, we apply calculus principles to derive the general solution for functions that integrate hyperbolic forms, showcasing the powerful tools calculus provides in advanced mathematics.