Problem 404 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

The antiderivative is \(\frac{1}{n}e^{nx} + C\).

Step 1: Understanding Hyperbolic Identities

Recall the hyperbolic identities: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \) and \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). Notice that \( \cosh(x) + \sinh(x) = e^x \). This simplification will make the integration process easier.

Step 2: Simplify the Function

Replace \( \cosh(x) + \sinh(x) \) with \( e^x \). Thus, the given function becomes \((e^x)^n = e^{nx}\).

Step 3: Integrate the Simplified Function

Find the antiderivative of \( e^{nx} \). The integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \), where \( C \) is the constant of integration.

Step 4: Apply Integration Formula

Using the formula from Step 3, the antiderivative of \( e^{nx} \) is \( \frac{1}{n} e^{nx} + C \). Make sure to include \( C \) to account for all possible antiderivatives.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the ordinary trigonometric functions but are based on hyperbolas instead of circles. The two primary hyperbolic functions are the hyperbolic sine and cosine, denoted as \( \sinh(x) \) and \( \cosh(x) \) respectively. These functions are defined as follows:

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

Hyperbolic functions share similar properties with trigonometric functions. For example, just like \( \cos^2 \theta + \sin^2 \theta = 1 \) for trigonometric functions, hyperbolic functions have the identity \( \cosh^2(x) - \sinh^2(x) = 1 \).
An interesting aspect of these functions is their relation to exponential functions. For instance, when simplifying \( \cosh(x) + \sinh(x) \), it results in \( e^x \). Understanding these identities can greatly simplify certain integration or differentiation problems, as it did in our exercise.

Integration Techniques

Integration techniques are a set of strategies used to find antiderivatives or integrals of functions. One of the foundational techniques is substitution, which can simplify a complex integral into a form that is easier to handle. For example, knowing the relationship between \( \cosh(x) + \sinh(x) \) and \( e^x \) allows us to use exponential integration techniques to handle our original problem.
The fundamental rule applied in the exercise involves the integration of exponential functions, such as \( e^{nx} \). Here, familiarity with integration rules for exponential functions is essential. The rule states that the integral of \( e^{ax} \) with respect to \( x \) is \( \frac{1}{a} e^{ax} + C \) where \( C \) is a constant of integration. Moreover, always remember to replace the original expression back after integrating the simplified form, ensuring the solution addresses the initial function presented in the problem statement.

Exponential Functions

Exponential functions are functions of the form \( f(x) = e^{ax} \), where \( e \) is Euler's number, a constant approximately equal to 2.71828. These functions are characterized by their rapid growth or decay and are used in various mathematical and real-world applications.
In mathematics, particularly calculus, exponential functions are known for their unique property: the derivative or integral of an exponential function is proportional to the function itself. For instance, the antiderivative of \( e^{nx} \) is \( \frac{1}{n} e^{nx} + C \), demonstrating how the process of integration directly reflects the form of the original function. To solve problems involving exponential functions using integration, substitution techniques that exploit these properties are often employed, transforming complex integrals to more manageable forms.