Problem 378 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

The antiderivatives are \(\sinh(x) + C\) for \(\cosh(x)\) and \(\cosh(x) + C\) for \(\sinh(x)\).

Step 1: Recall the Definitions of Hyperbolic Functions

The hyperbolic cosine function, \(\cosh(x)\), is defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\). Similarly, the hyperbolic sine function, \(\sinh(x)\), is defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\). We will use these definitions to find their respective antiderivatives.

Step 2: Compute the Antiderivative of \(\cosh(x)\)

To find the antiderivative of \(\cosh(x)\), we recognize that the derivative of \(\sinh(x)\) is \(\cosh(x)\). Therefore, the antiderivative of \(\cosh(x)\) is \(\sinh(x) + C\), where \(C\) is the constant of integration.

Step 3: Compute the Antiderivative of \(\sinh(x)\)

To find the antiderivative of \(\sinh(x)\), we realize that the derivative of \(\cosh(x)\) is \(\sinh(x)\). Thus, the antiderivative of \(\sinh(x)\) is \(\cosh(x) + C\), where \(C\) is the constant of integration.

Key Concepts

Hyperbolic Functions

In mathematics, hyperbolic functions are analogs of traditional trigonometric functions but for a hyperbola rather than a circle. They are essential in various fields like calculus and physics, helping in solving complex equations and representing hyperbolic curves.

There are two primary hyperbolic functions:

• Hyperbolic cosine, denoted as \(\cosh(x)\), which is defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).
• Hyperbolic sine, denoted as \(\sinh(x)\), defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).

These functions are related much like the sine and cosine functions in trigonometry, but with exponential growth characteristics. Hyperbolic functions are not periodic like the traditional trigonometric functions, which is crucial when analyzing real-world phenomena, especially those involving exponential growth or decay.

Understanding these functions' derivatives creates a foundation for finding their antiderivatives, making them an interesting aspect of calculus. For instance, the derivative of \(\sinh(x)\) is \(\cosh(x)\), and vice versa.

Integral Calculus

Integral calculus is a branch of mathematics concerned with finding functions from their rates of change or derivatives. It plays a crucial role in determining the accumulation of quantities and understanding the whole based on its parts. This is achieved through the concept of integration, which is essentially the reverse process of differentiation.

To integrate a function means finding an antiderivative or a function that differentiates back to the given function.

For example:

• The antiderivative of \(\cosh(x)\) is \(\sinh(x) + C\).
• The antiderivative of \(\sinh(x)\) is \(\cosh(x) + C\).

Here, each integration adds a constant \(C\), known as the constant of integration, which accounts for all possible functions that could have the same derivative.

This reflects how integral calculus allows flexibility and generality in finding functions that fit given derivative patterns, encompassing an infinite family of solutions based on initial conditions or additional constraints.

Constant of Integration

The constant of integration, often represented as \(C\), is a fundamental part of antiderivative solutions in integral calculus. When finding an antiderivative, we add this arbitrary constant \(C\) because derivative operations remove constant terms, making them "invisible" in the resulting derivative.

In the context of hyperbolic functions, when you find the antiderivative of \(\cosh(x)\) or \(\sinh(x)\), you'll say:

• For \(\cosh(x)\), the antiderivative is \(\sinh(x) + C\).
• For \(\sinh(x)\), the antiderivative is \(\cosh(x) + C\).

Adding \(C\) ensures that all potential solutions are covered since different original functions can share the same derivative.

This concept is crucial in solving indefinite integrals because it allows for

• A full representation of the family of all functions that satisfy the same differential equation.
• Multiple initial or boundary conditions can be accommodated, leading to a unique solution when needed.

Thus, the constant of integration is not just a mathematical formality but a necessary component ensuring completeness and accuracy in solving integration problems.