Problem 403 - Calculus Volume 1

Question

For the following exercises, find the antiderivatives for the given functions. $$\cosh (x)+\sinh (x)$$

Step-by-Step Solution

Answer

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

Step 1: Identify Basic Antiderivatives

Recall the basic antiderivatives for the hyperbolic functions. The antiderivative of \(\cosh(x)\) is \(\sinh(x) + C_1\), and the antiderivative of \(\sinh(x)\) is \(\cosh(x) + C_2\), where \(C_1\) and \(C_2\) are constants of integration.

Step 2: Apply Linearity of Integration

Due to the linearity property of integration, the antiderivative of \( \cosh(x) + \sinh(x) \) can be found by taking the antiderivative of each term separately.

Step 3: Compute the Antiderivatives

Using the results from Step 1, compute the antiderivative for each term: \( \int \cosh(x)\,dx = \sinh(x) + C_1\) and \( \int \sinh(x)\,dx = \cosh(x) + C_2\).

Step 4: Combine the Antiderivatives

Add the results from Step 3 to find the combined antiderivative: \( \sinh(x) + \cosh(x) + C \), where \(C = C_1 + C_2\) is the constant of integration.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles. These functions include hyperbolic sine, denoted as \( \sinh(x) \), and hyperbolic cosine, noted as \( \cosh(x) \). They are defined using exponential functions, making them essential in calculus for dealing with certain problems.

Here are the basic hyperbolic functions:

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

Unlike the periodic nature of sine and cosine, hyperbolic functions are not periodic. They have unique properties useful in various fields, such as engineering and physics.

For instance, \( \cosh(x) \) represents the shape of a hanging cable, also known as a catenary. Understanding hyperbolic functions helps in finding antiderivatives for functions similar to trigonometric ones.

Integration

Integration is a fundamental concept in calculus that helps find functions' areas underneath curves. It essentially reverses differentiation, bringing a function down from its derivative to the original or simpler form.

The process of finding an antiderivative or integral is vital for solving problems involving the area, acceleration, and other real-world applications. An integral's most prominent part is the constant of integration \( C \), showcasing the family of functions sharing the same derivative.

When integrating hyperbolic functions, recognize the basic antiderivatives:

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

Identifying these ensures that solving integrals involving hyperbolic functions becomes more accessible.

Linearity of Integration

Linearity of integration is a crucial property that enables integration to be straightforward and manageable. It allows us to break down the integration of a sum of functions into the sum of their respective integrals.

Mathematically, if you have two functions \( f(x) \) and \( g(x) \), then:
\[ \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx \]
This principle significantly simplifies the integration process. For our function \( \cosh(x) + \sinh(x) \), we applied linearity. We independently found the antiderivative for each function and then combined them. This yields an antiderivative of \( \sinh(x) + \cosh(x) + C \).

The linearity property is incredibly valuable. It means you can solve complex integration problems in smaller, more manageable steps.