Problem 400 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

\(\frac{1}{3} \tanh^3(x) + C\)

Step 1: Recall identity

Recall the hyperbolic identity \(\tanh^2(x) = 1 - \text{sech}^2(x)\). This might be helpful in simplifying the expression.

Step 2: Express tanh(x) in terms of sinh and cosh

Express the hyperbolic tangent in terms of hyperbolic sine and cosine: \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\). Thus, \(\tanh^2(x) = \frac{\sinh^2(x)}{\cosh^2(x)}\).

Step 3: Substitute into function

Substitute \(\tanh^2(x) = \frac{\sinh^2(x)}{\cosh^2(x)}\) into the original function: \(\tanh^2(x) \text{sech}^2(x) = \frac{\sinh^2(x)}{\cosh^2(x)} \cdot \frac{1}{\cosh^2(x)} = \frac{\sinh^2(x)}{\cosh^4(x)}\).

Step 4: Simplify the expression

Simplify the expression: \(\frac{\sinh^2(x)}{\cosh^4(x)}\) can be rewritten as \(\frac{1}{\cosh^2(x)} \left(\frac{\sinh(x)}{\cosh(x)}\right)^2 = \text{sech}^2(x) \tanh^2(x)\).

Step 5: Integration using substitution

Recognize \(\text{sech}^2(x)\) as the derivative of \(\tanh(x)\). Use substitution: let \(u = \tanh(x)\), then \(\frac{du}{dx} = \text{sech}^2(x)\). Thus, we integrate \(u^2\) with respect to \(u\), resulting in \(\int u^2 \, du = \frac{u^3}{3} + C\).

Step 6: Back substitution

Substitute back \(u = \tanh(x)\) into the integral result: \(\frac{\tanh^3(x)}{3} + C\).

Step 7: Present the final antiderivative

Therefore, the antiderivative of \(\tanh^2(x)\text{sech}^2(x)\) is \(\frac{1}{3}\tanh^3(x) + C\), where \(C\) is the constant of integration.

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric functions but are based on hyperbolas rather than circles. The primary hyperbolic functions include:

• Sinh (\(\sinh(x) = \frac{e^x - e^{-x}}{2}\)): Known as the hyperbolic sine.
• Cosh (\(\cosh(x) = \frac{e^x + e^{-x}}{2}\)): Known as the hyperbolic cosine.
• Tanh (\(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\)): Known as the hyperbolic tangent.
• Sech (\(\text{sech}(x) = \frac{1}{\cosh(x)}\)): Known as the hyperbolic secant.

Hyperbolic identities are similar to trigonometric identities. For instance, a crucial identity is \(\tanh^2(x) + \text{sech}^2(x) = 1\). This identity is helpful in simplifying expressions involving hyperbolic functions.

In the step-by-step solution, we utilized the \(\tanh^2(x) = 1 - \text{sech}^2(x)\) identity to reformulate the given function. This transformation is crucial for applying further integration techniques effectively.

Exploring the Substitution Method

The substitution method is a powerful integration technique used to simplify integrals. It involves introducing a new variable to make integration more straightforward. This is often done by identifying a part of the integrand whose derivative is also present. Here's how to effectively perform substitution:

• Identify a substitution variable: Look for a function within your integrand whose derivative is also present.
• Perform substitution: Replace the identified function and its differential in terms of the new variable.
• Integrate with respect to the new variable.
• Substitute back: After integration, substitute the original variable back.

In our example, we let \(u = \tanh(x)\) since \(\text{sech}^2(x)\) is the derivative of \(\tanh(x)\). This allowed the integration of \(u^2\) with respect to \(u\), resulting in \(\frac{u^3}{3} + C\). This simplifies the process and leads to a clean result.

Mastering Integration Techniques

Integration techniques refer to different methods used to find antiderivatives of functions. These techniques offer different approaches based on the complexity and form of the integrand. Some common integration techniques include:

• Substitution: Useful when the integrand resembles a derivative of a known function.
• Integration by Parts: Applicable when the integral can be broken into parts, using the formula \(\int u \cdot dv = uv - \int v \cdot du\).
• Partial Fraction Decomposition: Used for breaking down fractions into simpler parts before integrating.
• Trigonometric Integrals: Applicable when integrating products or powers of trigonometric functions.
• Hyperbolic Functions: As shown with \(\tanh^2(x)\text{sech}^2(x)\), hyperbolic functions can often be simplified using their identities.

Each technique has its own unique application. The choice of method often depends on the integrand's structure. In the exercise, recognizing the derivative within the integrand enabled us to efficiently apply substitution, simplifying the hyperbolic expression to find the antiderivative.