Question

For the following exercises, find the antiderivatives for the given functions.\(\tanh ^{2}(x) \operatorname{sech}^{2}(x)\)

Short answer

The antiderivative is \( \sinh(x) + C \).

Step-by-step solution

Identify the Function

The given function is \( \tanh^2(x) \operatorname{sech}^2(x) \). Our task is to find its antiderivative, which is also known as its indefinite integral.

Use Trigonometric Identity

Recall that \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \) and \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \). Therefore, \( \operatorname{sech}^2(x) = \frac{1}{\cosh^2(x)} \).

Simplify the Expression

Rewrite \( \tanh^2(x) \operatorname{sech}^2(x) \) as \( \left( \frac{\sinh(x)}{\cosh(x)} \right)^2 \cdot \frac{1}{\cosh^2(x)} \), which simplifies to \( \frac{\sinh^2(x)}{\cosh^4(x)} \).

Substitute and Integrate

Recognizing the derivative relationship, let \( u = \sinh(x) \), thus \( du = \cosh(x) dx \). Rewrite the integral in terms of \( u \): \( \int \frac{u^2}{u^4} du = \int 1 \cdot du = \int du \).

Integrate the Simplified Expression

The integral of \( du \) is simply \( u + C \), where \( C \) is the constant of integration. Thus, substituting back, we get \( \sinh(x) + C \).

Final Answer

The antiderivative of the function \( \tanh^2(x) \operatorname{sech}^2(x) \) is \( \sinh(x) + C \), where \( C \) is the constant of integration.

Key concepts

Trigonometric Identities

Trigonometric identities are very helpful in simplifying complex expressions involving trigonometric functions. These identities express one trigonometric function in terms of another. They're commonly used in calculus to simplify integrals and derivatives. For instance, in our exercise, the identity \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \) simplifies the task of integration. Similarly, the identity \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \) helps in expressing hyperbolic functions in a simpler form. Using these identities, you can rewrite or simplify expressions before performing calculus operations.
Breaking down the function \( \tanh^2(x) \operatorname{sech}^2(x) \) begins with these identities. It simplifies to \( \frac{\sinh^2(x)}{\cosh^4(x)} \), an easier form for integration. Trigonometric identities serve as a powerful tool in both solving problems and understanding the relationships between functions.

Indefinite Integrals

Indefinite integrals refer to the antiderivatives of a function. When you integrate a function without specific limits, the result is an expression that represents a family of functions. This is because the antiderivative includes a constant of integration, denoted as \( C \). In this way, integration is like reverse differentiation.
In this exercise, the task was to find the indefinite integral of \( \tanh^2(x) \operatorname{sech}^2(x) \). After simplifying the expression using trigonometric identities, we integrate the function with respect to \( x \). The goal is to find a function whose derivative equals the original function. Since integration essentially adds an infinite number of potential solutions, that's why we add the constant \( C \).
Thus, when you compute \( \int \frac{\sinh^2(x)}{\cosh^4(x)} \, dx \) and find it equals \( \sinh(x) + C \), the constant ensures that any vertical shift of the graph of the function will still solve the integral.

Hyperbolic Functions

Hyperbolic functions, like their trigonometric counterparts, have applications in different fields such as calculus and complex analysis. They include functions such as \( \sinh(x) \), \( \cosh(x) \), \( \tanh(x) \), and \( \operatorname{sech}(x) \). These functions mimic the behavior of ordinary trigonometric functions in the hyperbolic plane.
In this exercise, the functions \( \tanh(x) \) and \( \operatorname{sech}(x) \) play a major role. They resemble \( \tan(x) \) and \( \frac{1}{\cos(x)} \) respectively, but are suited for hyperbolic calculations. The identity \( \operatorname{sech}^2(x) = \frac{1}{\cosh^2(x)} \) assists in expressing the integrand in a more convenient form. By understanding these hyperbolic relationships, we can solve the integral by recognizing the function's derivative form in the calculation process.
Using hyperbolic identities simplifies calculus problems by providing alternative ways to express and manipulate expressions, making them essential in higher mathematics.