Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 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))
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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.
Recommended video:
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.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (1 + x²)
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)
Textbook Question
92. Evaluate ∫ from 3 to ∞ [ dx / (x √(x² - 9))]