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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.8.30
Chapter 8, Problem 8.8.30

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]

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1
Identify the integral to solve: \(\int_{2}^{4} \frac{dt}{t \sqrt{t^{2} - 4}}\).
Recognize that the integrand contains a square root of the form \(\sqrt{t^{2} - a^{2}}\), suggesting a trigonometric substitution. Use the substitution \(t = 2 \sec(\theta)\), where \(a = 2\).
Compute the differential \(dt\) in terms of \(d\theta\): since \(t = 2 \sec(\theta)\), then \(dt = 2 \sec(\theta) \tan(\theta) d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(t\), \(dt\), and simplifying the expression under the square root: \(\sqrt{t^{2} - 4} = \sqrt{4 \sec^{2}(\theta) - 4} = 2 \tan(\theta)\).
Change the limits of integration from \(t\) to \(\theta\) using \(t = 2 \sec(\theta)\), then simplify the integral and integrate with respect to \(\theta\).

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Key Concepts

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

Improper Integrals and Convergence

Understanding when an integral converges is crucial, especially for integrals with infinite limits or integrands with singularities. In this problem, the integral is definite and the integrand is continuous on [2,4], ensuring convergence. Recognizing convergence allows safe evaluation without concern for divergence.
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Improper Integrals: Infinite Intervals

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(t² − a²). By substituting t = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate. This method transforms the integral into a trigonometric integral that can be integrated directly.
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Introduction to Trigonometric Functions

Integration Techniques for Rational Functions

Integrals involving rational functions combined with radicals often require algebraic manipulation or substitution to simplify. Recognizing the structure of the integrand, such as t in the denominator and a radical in the denominator, guides the choice of substitution and integration steps to find an explicit antiderivative.
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Intro to Rational Functions
Related Practice
Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to ∞ of (dθ / (θ² - 1))

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Textbook Question

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

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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.

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Textbook Question

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))

Textbook Question

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 16x^3 (ln(x))^2 dx