Question

From the definitions of \(\cosh (x)\) and \(\sinh (x)\), find their antiderivatives.

Short answer

The antiderivative of \( \cosh(x) \) is \( \sinh(x) + C \), and the antiderivative of \( \sinh(x) \) is \( \cosh(x) + C \).

Step-by-step solution

Understanding Hyperbolic Functions

Identify the definitions of the hyperbolic functions. The hyperbolic cosine, \( \cosh(x) \), is defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). The hyperbolic sine, \( \sinh(x) \), is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

Determine the Antiderivative of \( \cosh(x) \)

The antiderivative of \( \cosh(x) \) can be directly found by integrating \( \cosh(x) \) with respect to \( x \). Look at the expression: \( \int \cosh(x) \, dx \). Since \( \sinh(x) \) is the derivative of \( \cosh(x) \), the antiderivative is \( \sinh(x) + C \) where \( C \) is the constant of integration.

Verify the Result for \( \cosh(x) \)

Let's verify by differentiating \( \sinh(x) \). We find \( \frac{d}{dx}(\sinh(x)) = \cosh(x) \), which confirms that the antiderivative \( \sinh(x) + C \) is correct.

Determine the Antiderivative of \( \sinh(x) \)

Integrate \( \sinh(x) \) by observing the expression: \( \int \sinh(x) \, dx \). Since \( \cosh(x) \) is the derivative of \( \sinh(x) \), the antiderivative is \( \cosh(x) + C \).

Verify the Result for \( \sinh(x) \)

Check the antiderivative by differentiating \( \cosh(x) \). We have \( \frac{d}{dx}(\cosh(x)) = \sinh(x) \). This verification shows the antiderivative \( \cosh(x) + C \) is indeed correct.

Key concepts

Antiderivatives

Antiderivatives are a fundamental concept in calculus, closely linked with integration. Imagine reversing the process of differentiation; that's what finding an antiderivative means. If you're given a function, its antiderivative is another function whose derivative brings you back to the original. Think of it like finding the original stretch of film given just a clip. It's all about understanding what happened before.

To illustrate, consider the hyperbolic functions like \( \cosh(x) \) and \( \sinh(x) \). Each has a counterpart, known as its derivative. Thus, the antiderivative of \( \cosh(x) \) is \( \sinh(x) + C \), and vice versa for \( \sinh(x) \). Here, \( C \) represents an arbitrary constant, accounting for perfect generalization.

Integration

Integration is one of the cornerstones of calculus, providing the means to accumulate quantities or reverse differentiation. When you integrate a function, you're essentially finding its antiderivative. It's a method for summing infinitesimally small slices to find the whole. Just like piecing together a puzzle to see the full picture.

For hyperbolic functions, integrating \( \cosh(x) \) results in \( \sinh(x) + C \), and integrating \( \sinh(x) \) gives \( \cosh(x) + C \). Here, integration confirms the relationship between these particular hyperbolic functions and reinforces their connection as derivatives to one another. Always remember, after integration, an arbitrary constant \( C \) should be included to represent the general solution.

Hyperbolic Sine

Hyperbolic sine, denoted \( \sinh(x) \), is a vital function in the world of hyperbolic functions. Defined by \( \sinh(x) = \frac{e^x - e^{-x}}{2} \), it reflects the unique properties of exponential functions. Imagine it as analogous to the sine function in trigonometry, yet connected to exponential growth and decay instead.

In terms of calculus, \( \sinh(x) \) serves as the antiderivative of \( \cosh(x) \). When differentiating \( \sinh(x) \), you regain \( \cosh(x) \), showcasing their tightly knit relationship. Similarly, integrating \( \sinh(x) \) leads you to \( \cosh(x) + C \). These operations underline how hyperbolic sine functions harmonize within the calculus framework.

Hyperbolic Cosine

Approach hyperbolic cosine, denoted as \( \cosh(x) \), with intrigue, as it mirrors the cosine function's attributes in trigonometry through a hyperbolic lens. Defined by \( \cosh(x) = \frac{e^x + e^{-x}}{2} \), this function carries the elegance of symmetry and growth, similar yet unique from the circular counterparts.

In calculus, integrating \( \cosh(x) \) results in \( \sinh(x) + C \), and differentiating \( \cosh(x) \) leads you back to \( \sinh(x) \). These procedures reveal how \( \cosh(x) \) serves as both the integral of \( \sinh(x) \) and its derivative, nicely tying their roles together. It’s essential to appreciate \( \cosh(x) \) as a cornerstone in the study of hyperbolic functions, offering depth to mathematical concepts exploring symmetry, growth, and smooth curves.