Problem 49 - Calculus of a Single Variable

Question

In Exercises 43–54, find the indefinite integral. $$ \int \frac{\cosh x}{\sinh x} d x $$

Step-by-Step Solution

Answer

\(\int \frac{\cosh x}{\sinh x} dx = ln|sinh x| + C\)

Step 1: Make Use of the Reciprocal Identity

Recall that the reciprocal of sinh x is coth x, therefore we can rewrite the expression as \(\int coth x dx\) and simplify the integration.

Step 2: Perform the Integration

The integral of coth x is defined as ln|sinh x|. Hence, \(\int coth x dx = ln|sinh x|\).

Step 3: Write the Final Answer

It's important to remember to add the constant of integration, C, to the final answer, as we are performing indefinite integration. Therefore, the final answer is \(ln|sinh x| + C\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogues to the ordinary trigonometric functions, but for a hyperbola, as trigonometric functions relate to the circle. Just as trigonometric functions are essential in studying circles and periodic phenomena, hyperbolic functions are important when dealing with hyperbolas and certain types of growth patterns and waves.

There are two primary hyperbolic functions: the hyperbolic sine function, often denoted as \(\sinh x\), and the hyperbolic cosine function, denoted as \(\cosh x\). These functions are defined in terms of exponential functions as follows:
\[\sinh x = \frac{e^x - e^{-x}}{2}\]and
\[\cosh x = \frac{e^x + e^{-x}}{2}\]
There are also other hyperbolic functions, such as hyperbolic tangent, denoted as \(\tanh x\), and hyperbolic cotangent \(\coth x\), both derived from the primary hyperbolic functions. The functions appear in various mathematical contexts like solving certain differential equations and modeling physical phenomena.

For example, in our exercise, we focus on the integration of a hyperbolic function. Intuition from trigonometry can often be adapted to hyperbolic functions, such as understanding their properties and behavior, which can aid in solving integrals that involve them.

Integration Techniques

Integration is a fundamental technique in calculus, used to find areas under curves, amongst many other applications. Several integration techniques are crucial to solving indefinite integrals, which, unlike definite integrals, do not have limits of integration and result in a general form plus a constant of integration, C.

Common integration techniques include:

• Substitution, which involves changing variables to simplify an integral.
• Integration by parts, which is based on the product rule for differentiation.
• Partial fractions, which breaks down complex rational expressions into simpler ones.
• Trigonometric substitution, which is useful for integrals involving square roots of quadratic polynomials.

Understanding when and how to apply these techniques is critical for solving more complex integration problems. In our original problem, we didn't have to utilize any sophisticated methods because the integral presented could be simplified through the knowledge of hyperbolic function identities. Taking advantage of these identities often simplifies the integration process, as was the case in the step-by-step solution provided.

Reciprocal Identities

Reciprocal identities are a set of equations that express the relationship between a function and its reciprocal. In the context of trigonometry, these identities are used to simplify expressions and solve equations. Similarly, hyperbolic functions have their own reciprocal identities.

For example, the reciprocal of \(\sinh x\) is \(\coth x\), which is the hyperbolic cotangent function. This is written as:\[\coth x = \frac{1}{\sinh x} = \frac{\cosh x}{\sinh x}\]
The reciprocal identities are particularly useful when integrating, as they allow us to transform the integral into a form that is more recognizable and easier to solve. In the exercise at hand, recognizing that \(\cosh x/\text{sinh} x\) is the same as \(\coth x\) was crucial to simplifying the integral. Instead of a more complex integration process, the problem was reduced to integrating a known reciprocal identity, which is a straightforward task and demonstrates the power of understanding these mathematical relationships.