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Chapter 6: Problem 35

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

Short Answer

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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 by step solution

01

Recall the Pythagorean Identity for Tanh

Remember that the identity for hyperbolic tangent function, \(\tanh{x}\), is given as: \(\tanh^2{x} = 1 - \text{sech}^2{x}\)
02

Rewrite the given integral using the Identity

Replace \(\tanh^2{x}\) in the given integral with \(1 - \text{sech}^2{x}\). Our integral becomes: \(\int (\tanh^2{x})\,dx = \int (1 - \text{sech}^2{x})\,dx\)
03

Break down the integral

Break down the integral into two separate integrals: \(\int (1 - \text{sech}^2{x})\,dx = \int 1\,dx - \int \text{sech}^2{x}\,dx\)
04

Integrate 1 with respect to x

Integrate the first part, \(\int 1\,dx\), using the basic integration rule: \(\int 1\,dx = x + C_1\)
05

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: \(\int \text{sech}^2{x}\,dx = \tanh x + C_2\)
06

Combine the results

Combine the results from the two separate integrals: \(\int 1\,dx - \int \text{sech}^2{x}\,dx = (x + C_1) - (\tanh x + C_2)\)
07

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places. a. \(\coth 4\) b. \(\tanh ^{-1} 2\) c. \(\operatorname{csch}^{-1} 5\) d. \(\left.\operatorname{csch} x\right|_{1 / 2} ^{2}\) e. \(\left.\ln \left|\tanh \left(\frac{x}{2}\right)\right|\right|_{1} ^{10}\) f. \(\left.\tan ^{-1}(\sinh x)\right|_{-3} ^{3}\) g. \(\left.\frac{1}{4} \operatorname{coth}^{-1} \frac{x}{4}\right|_{20} ^{36}\)

Kelly started at noon \((t=0)\) riding a bike from Niwot to Berthoud, a distance of \(20 \mathrm{km},\) with velocity \(v(t)=15 /(t+1)^{2}\) (decreasing because of fatigue). Sandy started at noon \((t=0)\) riding a bike in the opposite direction from Berthoud to Niwot with velocity \(u(t)=20 /(t+1)^{2}\) (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours. a. Make a graph of Kelly's distance from Niwot as a function of time. b. Make a graph of Sandy's distance from Berthoud as a function of time. c. When do they meet? How far has each person traveled when they meet? d. More generally, if the riders' speeds are \(v(t)=A /(t+1)^{2}\) and \(u(t)=B /(t+1)^{2}\) and the distance between the towns is \(\vec{D},\) what conditions on \(A, B,\) and \(D\) must be met to ensure that the riders will pass each other? e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

There are several ways to express the indefinite integral of \(\operatorname{sech} x\). a. Show that \(\left.\int \operatorname{sech} x d x=\tan ^{-1}(\sinh x)+C \text { (Theorem } 6.9\right)\) (Hint: Write sech \(x=\frac{1}{\cosh x}=\frac{\cosh x}{\cosh ^{2} x}=\frac{\cosh x}{1+\sinh ^{2} x}\) and then make a change of variables.) b. Show that \(\int \operatorname{sech} x d x=\sin ^{-1}(\tanh x)+C .\) (Hint: Show that \(\operatorname{sech} x=\frac{\operatorname{sech}^{2} x}{\sqrt{1-\tanh ^{2} x}}\) and then make a change of variables.) c. Verify that \(\int \operatorname{sech} x d x=2 \tan ^{-1} e^{x}+C\) by proving \(\frac{d}{d x}\left(2 \tan ^{-1} e^{x}\right)=\operatorname{sech} x.\)

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of \(5 \mathrm{kg} / \mathrm{m}\). a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain onto the cylinder if a \(50-\mathrm{kg}\) block is attached to the end of the chain?

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=\frac{1}{\sqrt{x^{2}+1}}\) and \(y=\frac{1}{\sqrt{2}}\) revolved about the \(x\) -axis
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