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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.6.69
Chapter 7, Problem 7.6.69

Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))

Verified step by step guidance
1
Identify the integral to solve: \(\int \frac{dx}{(2x-1) \sqrt{(2x-1)^2 - 4}}\).
Make a substitution to simplify the integral. Let \(u = 2x - 1\), so that \(du = 2 \, dx\) or \(dx = \frac{du}{2}\).
Rewrite the integral in terms of \(u\): \(\int \frac{\frac{du}{2}}{u \sqrt{u^2 - 4}} = \frac{1}{2} \int \frac{du}{u \sqrt{u^2 - 4}}\).
Recognize the integral form \(\int \frac{du}{u \sqrt{u^2 - a^2}}\), which suggests using a trigonometric substitution such as \(u = 2 \sec \theta\) to simplify the square root.
Perform the substitution \(u = 2 \sec \theta\), find \(du\), and rewrite the integral in terms of \(\theta\) to proceed with integration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration involving square roots of quadratic expressions

Integrals containing expressions like √(ax + b)² - c² often require recognizing the form as a difference of squares under the root. This suggests using trigonometric or hyperbolic substitutions to simplify the integral by transforming the radical into a simpler expression.
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Integrals Involving Natural Logs: Substitution Example 7

Trigonometric substitution

Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving radicals such as √(x² - a²). For example, setting x = a sec θ transforms the radical into a trigonometric expression, making the integral easier to evaluate.
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Introduction to Trigonometric Functions

Substitution method (u-substitution)

U-substitution involves changing variables to simplify the integral, especially when the integrand contains a composite function. Identifying a part of the integrand as u and expressing dx in terms of du can reduce the integral to a basic form that is easier to integrate.
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Substitution With an Extra Variable
Related Practice
Textbook Question

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.


f(x) = 1/x², x > 0

Textbook Question

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

63. tanh⁻¹(-1/2)

Textbook Question

Evaluate the integrals in Exercises 39–56.

41. ∫2y dy/(y²-25)

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Textbook Question

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = ln(e^(θ)/(1+e^θ))

Textbook Question

Evaluate the integrals in Exercises 39–56.

52. ∫(from π/4 to π/2)cot(t)dt

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Textbook Question

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

31. y = cos⁻¹(x) - x sech⁻¹(x)