Problem 399 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

We need to find the antiderivative of the function \( \cosh^{2}(x) \sinh(x) \). This function involves hyperbolic trigonometric functions, specifically \( \cosh(x) \) and \( \sinh(x) \).

Step 2: Choose a Substitution

Notice that the function \( \cosh^{2}(x) \sinh(x) \) can be simplified using substitution. Let \( u = \cosh(x) \), then the derivative \( \frac{du}{dx} = \sinh(x) \), or \( du = \sinh(x) \, dx \).

Step 3: Rewrite the Integral

With the substitution \( u = \cosh(x) \), rewrite the integral as follows: \( \int \cosh^{2}(x) \sinh(x) \, dx = \int u^{2} \, du \).

Step 4: Integrate with Respect to \( u \)

Now, find the antiderivative of \( u^{2} \, du \), which is \( \frac{u^{3}}{3} + C \), where \( C \) is the constant of integration.

Step 5: Back-Substitute to Original Variable

Substitute back \( u = \cosh(x) \) into the antiderivative to return to the original variable: \( \frac{\cosh^{3}(x)}{3} + C \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions that are similar to trigonometric functions but relate to hyperbolas instead of circles. They include functions like the hyperbolic sine (\( \sinh(x) \)) and hyperbolic cosine (\( \cosh(x) \)). Understanding these functions is crucial when working with problems involving hyperbolic identities or solving integrals that involve them.
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Here's a quick overview to remember:

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

These functions have properties that make them useful in solving differential equations, performing transformations, and in physics, particularly in general relativity. Similar to trigonometric identities, hyperbolic functions also satisfy certain identities, such as \( \cosh^2(x) - \sinh^2(x) = 1 \), which can be very useful in calculus.

Substitution Method

The substitution method is a crucial and common technique in integration, especially when dealing with more complex functions. In essence, it involves changing variables to transform a difficult integral into a simpler one.
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To use substitution, you typically:

• Select a portion of the integrand to substitute with a new variable \( u \).
• Differentiate to find \( du \).
• Rewrite the integral in terms of \( u \) and \( du \).

This is exactly what was done in the original exercise:

• We identified \( u = \cosh(x) \) and \( du = \sinh(x) \, dx \).
• The integral transformed from \( \int \cosh^2(x) \sinh(x) \, dx \) to \( \int u^2 \, du \).

By substituting, we simplified the integral to one which is straightforward to solve. This method is particularly powerful because it doesn't just apply to hyperbolic functions but to a wide range of integrals.

Integration Techniques

Integration techniques are essential tools for finding antiderivatives of functions, and using the right technique can greatly simplify your work. In the original exercise, with the help of substitution, we applied a basic power rule of integration.
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Once in a simpler form, we integrated \( u^2 \) (a polynomial term) easily using the power rule:

• The integral of \( u^n \) is \( \frac{u^{n+1}}{n+1} + C \), as long as \( n eq -1 \).
• So, for \( \int u^2 \, du \), we applied \( \int u^2 \, du = \frac{u^3}{3} + C \).

In addition to substitution and power rule, many other integration techniques can be employed, such as:

• Integration by Parts
• Partial Fraction Decomposition
• Trigonometric Identities

Selecting the appropriate technique depends on the function you're working with. A good understanding of these techniques can significantly aid in solving complex calculus problems.