Question

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

Short answer

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

Step-by-step solution

Recognize the Given Function

The problem requires us to find the antiderivative of the function \((\cosh(x) + \sinh(x))^n\). In mathematics, finding the antiderivative is equivalent to finding the indefinite integral of the given function.

Use a Trigonometric Identity

Notice that \(\cosh(x) + \sinh(x)\) simplifies using the identity: \(\cosh(x) = \frac{e^x + e^{-x}}{2}\) and \(\sinh(x) = \frac{e^x - e^{-x}}{2}\). Thus:\[\cosh(x) + \sinh(x) = e^x\]

Substitute and Simplify

Substituting \(e^x\) for \(\cosh(x) + \sinh(x)\), the function transforms into \((e^x)^n = e^{nx}.\) This reduces the problem to finding the antiderivative of \(e^{nx}\).

Find the Antiderivative

The antiderivative of \(e^{nx}\) with respect to \(x\) is found using the formula:\[\int e^{nx} \, dx = \frac{1}{n} e^{nx} + C\]Here, \(C\) is the constant of integration.

Write the Final Answer

Thus, the antiderivative of \((\cosh(x) + \sinh(x))^n\) is \(\frac{1}{n} e^{nx} + C\).

Key concepts

Indefinite Integral

The indefinite integral, often represented as \( \int f(x) \, dx \), is the process of finding a function whose derivative is the given function \( f(x) \). This is often referred to as finding the antiderivative. One key aspect of indefinite integrals is that they include a constant of integration, denoted by \( C \). This is because the derivative of any constant is zero, so we cannot determine from the derivative alone the exact constant that was present in the original function.
To tackle indefinite integrals, one must often be familiar with a variety of integration rules such as power rule, exponential rule, and more. These rules help transform the given function into a form that is easier to integrate. Indefinite integrals are crucial in many fields such as physics, engineering, and economics, where they help in calculating quantities like displacement, area, and total income over time.

Exponential Functions

Exponential functions are functions of the form \( f(x) = a \, e^{kx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions are notable for their rapid growth and are used widely in compound interest problems, population growth models, and natural processes like radioactive decay.
When integrating exponential functions, the process often involves recognizing that the antiderivative of \( e^{kx} \) is \( \frac{1}{k} \, e^{kx} \), where \( k \) is a constant. This arises from the fundamental property that the derivative and integral of \( e^x \) are intrinsically tied to the base \( e \), making these calculations quite distinct compared to other function types.
Comprehending exponential functions and their integration properties is pivotal, as they simplify into an easy and elegant form after integration, which can be seen in problems involving exponential growth or decay.

Hyperbolic Functions

Hyperbolic functions, expressed as \( \sinh(x) \) and \( \cosh(x) \), are analogs of the trigonometric functions but for the hyperbola, much like sine and cosine are for the circle. These functions are defined as follows:

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

Hyperbolic functions are used in many fields, including hyperbolic geometry, modeling high-speed motion in physics, and describing certain electrical circuits.
When working with hyperbolic functions in integration problems, it can be beneficial to use their exponential definitions. This transforms complex hyperbolic integrands into simpler forms, often involving exponential functions, which are typically more straightforward to integrate. Understanding these relationships can greatly facilitate solving integration problems involving hyperbolic functions.

Integration Techniques

Integration techniques are methods used to evaluate integrals, especially when the function does not easily lend itself to straightforward integration. To solve the given problem, one crucial technique employed was substitution, specifically using trigonometric and hyperbolic identities.
In more complex problems, other integration techniques such as parts, partial fraction decomposition, or trigonometric substitution might be necessary. Each technique is designed to handle specific forms of integrands:

• **Substitution**: Useful when the function can be rewritten in a simpler form.
• **Integration by Parts**: Applies to products of functions, using the formula \( \int u \, dv = uv - \int v \, du \).
• **Partial Fraction Decomposition**: Useful for rational functions, breaking them into simpler fractions.

Mastering these techniques allows one to tackle a wide variety of integration problems effectively, significantly easing the process of finding antiderivatives for complex functions.