Problem 35 - Calculus Early Transcendentals

Question

Determine each indefinite integral. \(\int \tanh ^{2} x d x\) (Hint: Use an identity.)

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the hyperbolic tangent squared function, \(\int (\tanh^2{x})\,dx\). Answer: The indefinite integral of the hyperbolic tangent squared function is given by \(\int (\tanh^2{x})\,dx = x - \tanh x + C\), where \(C\) is the constant of integration.

Step 1: Recall the Pythagorean Identity for Tanh

Remember that the identity for hyperbolic tangent function, \(\tanh{x}\), is given as: \(\tanh^2{x} = 1 - \text{sech}^2{x}\)

Step 2: Rewrite the given integral using the Identity

Replace \(\tanh^2{x}\) in the given integral with \(1 - \text{sech}^2{x}\). Our integral becomes: \(\int (\tanh^2{x})\,dx = \int (1 - \text{sech}^2{x})\,dx\)

Step 3: Break down the integral

Break down the integral into two separate integrals: \(\int (1 - \text{sech}^2{x})\,dx = \int 1\,dx - \int \text{sech}^2{x}\,dx\)

Step 4: Integrate 1 with respect to x

Integrate the first part, \(\int 1\,dx\), using the basic integration rule: \(\int 1\,dx = x + C_1\)

Step 5: Integrate sech^2{x} with respect to x

Now, we need to integrate the second part, \(\int \text{sech}^2{x}\,dx\). Recall the property that the derivative of hyperbolic tangent is given by the square of hyperbolic secant function: \(\frac{d \tanh x}{d x} = \text{sech}^2{x}\) So, the indefinite integral of \(\text{sech}^2{x}\) is: \(\int \text{sech}^2{x}\,dx = \tanh x + C_2\)

Step 6: Combine the results

Combine the results from the two separate integrals: \(\int 1\,dx - \int \text{sech}^2{x}\,dx = (x + C_1) - (\tanh x + C_2)\)

Step 7: Write the final result

Simplify the expression and write the final result as a combined constant \(C\): \(\int (\tanh^2{x})\,dx = x - \tanh x + C\) Where \(C = (C_1 - C_2)\) is the constant of integration.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, much like their trigonometric counterparts, are essential in calculus and other branches of mathematics. They are linked to the exponential function and are used to model real-world phenomena such as waves and heat transfer.

• The basic hyperbolic functions include the hyperbolic sine, \(\sinh(x)\), and the hyperbolic cosine, \(\cosh(x)\).
• From these, other functions like hyperbolic tangent, \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\), are derived.
• Each hyperbolic function has an identity, similar to trigonometric identities.

Understanding these functions is crucial as they offer convenient solutions through identities for integration and differentiation, particularly in solving differential equations.

Integration Techniques

Integration techniques are essential tools to solve indefinite integrals. These methods transform complex expressions into simpler forms, making them easier to integrate. In our example, we use a basic strategy: direct integration and substitution.

• Begin by recognizing identities that can simplify the integrand. For example, the identity \(\tanh^2{x} = 1 - \text{sech}^2{x}\) allows us to break down the integral.
• Divide the integral into simpler parts, such as separating \(\int (1 - \text{sech}^2{x})\,dx\) into two individual integrals.
• Tackle each part separately by applying direct integration for constants and recognizing standard derivatives, like \(\int \text{sech}^2{x} \,dx = \tanh{x}\).

These techniques are foundational in solving more complex integrals and prepare students for advanced calculus topics.

Pythagorean Identities

Pythagorean identities provide powerful tools to relate different functions, simplifying integrals and derivatives. For hyperbolic functions, similar identities help break down complex expressions.

• The identity used here is \(\tanh^2{x} = 1 - \text{sech}^2{x}\), mirroring the trigonometric identity \(\sin^2{x} + \cos^2{x} = 1\).
• This identity allows us to rewrite the original integral \(\int \tanh^2{x} \,dx\) into components that are easier to integrate.
• It highlights how understanding these identities is critical for simplifying expressions and finding exact solutions in calculus.

Mastering Pythagorean identities aids in tackling both hyperbolic and trigonometric integrals effectively.