Problem 48 - Calculus of a Single Variable

Question

In Exercises 43–54, find the indefinite integral. $$ \int \frac{\sinh x}{1+\sinh ^{2} x} d x $$

Step-by-Step Solution

Answer

The integral yields \( \ln |\sinh x|+ C \)

Step 1: Appropriate Substitution

To simplify the integral, a change in variable can be done with the substitution \( \sinh x = t \). Calculating derivative wrt to \(x\), \( \cosh x dx = dt \). Now, the given formula can be rewritten utilizing the substitution \( t = \sinh x \) and the differential \( dt = \cosh x dx \)

Step 2: Simplification using Formulas

The term \(1+\sinh ^{2} x\) can be simplified using the formula \(\cosh^2 x = 1 + \sinh^2 x\). So the integral simplifies to \( \int \frac{t}{ \cosh^2 x} dt \). Using the substitution \( dt = \cosh x dx \), the integral becomes \( \int \frac{1}{t} dt \)

Step 3: Final Integration

The integral \( \int \frac{1}{t} dt \) is a standard integral whose value is \( \ln |t| + C \), where \(C\) is the constant of integration.

Step 4: Back-substituting the original variable

Substituting back the original variable, replacing \( t = \sinh x \), the final answer is \( \ln |\sinh x|+ C \)

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. Just as trigonometric functions are defined in terms of the unit circle, hyperbolic functions are defined in terms of a hyperbola. The two main hyperbolic functions are the hyperbolic sine (\text{sinh}) and hyperbolic cosine (\text{cosh}), which are defined by the exponential functions:

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

Other hyperbolic functions like hyperbolic tangent (\text{tanh}) and hyperbolic cotangent (\text{coth}) are derived from these. One of the key properties of hyperbolic functions is the identity \text{cosh}^2(x) - \text{sinh}^2(x) = 1, mirroring the Pythagorean identity \text{sin}^2(x) + \text{cos}^2(x) = 1 for trigonometric functions. This relation helps in integrating functions involving hyperbolic functions by substituting hyperbolic identities. For example, in the indefinite integral exercise \[ \int \frac{\text{sinh} x}{1+\text{sinh} ^{2} x} \text{dx} \] knowing the identity \text{cosh}^2(x) - \text{sinh}^2(x) = 1 is useful to simplify the expression in the denominator.

Integration by Substitution

Integration by substitution, also known as u-substitution, is a method used to find the integral of a function that involves a composite function. The basic idea is to simplify the integral by substituting a part of the integrand with a new variable. This involves two main steps:

• Choosing a substitution that simplifies the integrand.
• Finding the differential of the new variable and substituting it for the original differential in the integral.

For example, in the integral \[ \int \frac{\text{sinh} x}{1+\text{sinh} ^{2} x} \text{dx} \] the substitution \text{sinh}(x) = t simplifies the integrand into a form that is easier to integrate. After substituting, it's crucial to also change the differential \text{dx} to \text{dt} based on the derivative of the chosen substitution. The success of this method largely depends on choosing a substitution that makes the integral simpler and often involves recognizing the patterns that match standard integral forms.

Trigonometric Substitution

Trigonometric substitution is a technique for evaluating integrals that allows us to substitute trigonometric functions for algebraic expressions. This method is particularly effective for integrals that involve square roots of quadratic expressions or that resemble the forms of Pythagorean identities. The process involves:

• Identifying an expression within the integral that matches a trigonometric identity.
• Using the identified expression to define a trigonometric substitution.
• Expressing the integral in terms of the trigonometric function and simplifying.

While the exercise given involves hyperbolic functions and uses integration by substitution, it's insightful to know the parallels with trigonometric substitution. Similar strategies may apply, such as looking for identities or expressions that simplify the integrand. However, it's important to remember that hyperbolic function identities differ from trigonometric identities, although they may look similar in structure. Using the correct identity is key to finding the right substitution and successfully integrating the function.