Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
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Ch. 8 - Techniques of Integration
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 8, Problema 8.6.34
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ dt / (tan(t)√4 - sin^2(t))
Guida verificata passo dopo passo1
Identify the integral: \(\int \frac{dt}{\tan(t) \sqrt{4 - \sin^2(t)}}\).
Rewrite the integral in terms of sine and cosine to simplify: recall that \(\tan(t) = \frac{\sin(t)}{\cos(t)}\), so the integral becomes \(\int \frac{dt}{\frac{\sin(t)}{\cos(t)} \sqrt{4 - \sin^2(t)}} = \int \frac{\cos(t)}{\sin(t) \sqrt{4 - \sin^2(t)}} dt\).
Use the substitution \(u = \sin(t)\), which implies \(du = \cos(t) dt\). This substitution will help rewrite the integral in terms of \(u\).
Rewrite the integral in terms of \(u\): since \(\cos(t) dt = du\), the integral becomes \(\int \frac{du}{u \sqrt{4 - u^2}}\).
Recognize that the integral \(\int \frac{du}{u \sqrt{4 - u^2}}\) is a standard form that can be found in integral tables or solved using further substitution or partial fractions.
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Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving square roots of expressions like a² - x², a² + x², or x² - a². By substituting a trigonometric function for the variable, the integral transforms into a form involving trigonometric identities, making it easier to evaluate.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Integration Using Tables
Integration tables provide standard integral forms and their solutions, allowing quick evaluation of complex integrals once transformed appropriately. After substitution, matching the integral to a known form in the table helps in directly writing down the antiderivative without performing the integration from scratch.
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Percorso guidato
08:01Integration Using Partial Fractions
Manipulation of Trigonometric Functions
Understanding the relationships and identities between trigonometric functions like sine, cosine, and tangent is essential. Simplifying expressions such as tan(t) and √(4 - sin²(t)) often requires using identities like sin²(t) + cos²(t) = 1, which aids in rewriting the integral into a more manageable form.
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Percorso guidato
6:04Introduction to Trigonometric Functions
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