Domanda del libro di testo
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.6.69
Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))
Guida verificata passo dopo passo1
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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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.
Video consigliato:
03:33
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.
Video consigliato:
Percorso guidato
6:04Introduction 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.
Video consigliato:
04:27Substitution With an Extra Variable
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In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
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