Question

For the following exercises, find the antiderivatives for the given functions.\(\cosh ^{2}(x) \sinh (x)\)

Short answer

The antiderivative is \( \frac{\cosh^3(x)}{3} + C \).

Step-by-step solution

Identify the Integral to Solve

The task is to find the antiderivative of the function \(\cosh^2(x) \sinh(x)\). This is equivalent to integrating the function with respect to \(x\): \[ \int \cosh^2(x) \sinh(x) \, dx \]

Use Substitution to Simplify

To simplify the integration process, use the substitution method. Let \( u = \cosh(x) \). Then, differentiate \( u \) with respect to \( x \) to find \( du \): \[ du = \sinh(x) \, dx \] Substitute \( u \) and \( du \) into the integral:\[ \int u^2 \, du \]

Integrate the Simplified Expression

Now, integrate \( u^2 \) with respect to \( u \):\[ \int u^2 \, du = \frac{u^3}{3} + C \] where \( C \) is the constant of integration.

Substitute Back the Original Variable

Revert back to the original variable \( x \) by substituting \( u = \cosh(x) \) into the integrated result:\[ \frac{(\cosh(x))^3}{3} + C \] Thus, the antiderivative of \( \cosh^2(x) \sinh(x) \) is\[ \frac{\cosh^3(x)}{3} + C \]

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs to the trigonometric functions we commonly know, but they are based around the hyperbola rather than the circle. The most common hyperbolic functions are the hyperbolic sine, denoted as \( \sinh(x) \), and the hyperbolic cosine, denoted as \( \cosh(x) \). These functions have profound applications in various fields like calculus, physics, and engineering.

To understand these functions better, you can relate them to their exponential form:

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

This representation makes certain mathematical manipulations easier, especially in calculus where hyperbolic identities are used. For instance, \( \cosh^2(x) - \sinh^2(x) = 1 \) mirrors the Pythagorean identity for sine and cosine. Similar identities help in integrating and differentiating expressions involving hyperbolic functions.

Substitution Method

The substitution method is a powerful technique in calculus used to simplify integrals by introducing a new variable. This can transform a difficult or complex integral into a more manageable form.
To apply this method, follow these steps:

• Select a substitution: In the given problem, we substitute \( u = \cosh(x) \).
• Differentiate the substitution: Find \( du \) in terms of \( dx \). In this case, \( du = \sinh(x) \, dx \).
• Rewrite the integral: Replace all instances of the original variables with the new variable \( u \). Here, \( \int \cosh^2(x) \sinh(x) \, dx \) becomes \( \int u^2 \, du \).

By simplifying the integral, substitution allows for easier evaluation. This technique is widely used and is especially handy when dealing with compositions of functions.

Integration Techniques

Integration techniques provide a toolkit for finding antiderivatives of various functions. Beyond basic rules, these techniques include substitution, integration by parts, and partial fractions.

Here, we used a basic substitution technique to evaluate \( \int u^2 \, du \). This resulted in integrating a simple polynomial:

• The antiderivative of \( u^2 \) is \( \frac{u^3}{3} + C \), where \( C \) is the integration constant.

Once the integration is completed in terms of the new variable \( u \), it is important to substitute back to the original variable. This ensures that the solution is expressed in the terms initially given in the problem.

Mastering these techniques allows one to tackle a wide range of integration problems, making them invaluable tools in both calculus and beyond.