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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 8.4.2
Chapter 8, Problem 8.4.2

Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)

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1
Recognize that the integral has the form \(\int \frac{3 \, dx}{\sqrt{1 + 9x^2}}\), which resembles the standard integral \(\int \frac{dx}{\sqrt{a^2 + x^2}}\) whose antiderivative involves a hyperbolic or inverse hyperbolic function or a logarithm depending on the substitution.
Identify the constant inside the square root: here, \(a^2 = 1\) and the term with \(x^2\) is \(9x^2 = (3x)^2\). This suggests using a substitution to simplify the expression under the square root.
Make the substitution \(u = 3x\), which implies \(du = 3 \, dx\) or equivalently \(dx = \frac{du}{3}\). This will help rewrite the integral in terms of \(u\).
Rewrite the integral in terms of \(u\): substitute \(3 \, dx = du\) and \(\sqrt{1 + 9x^2} = \sqrt{1 + u^2}\), so the integral becomes \(\int \frac{du}{\sqrt{1 + u^2}}\).
Recall the antiderivative formula \(\int \frac{du}{\sqrt{1 + u^2}} = \sinh^{-1}(u) + C\) or equivalently \(\ln|u + \sqrt{1 + u^2}| + C\). After integrating, substitute back \(u = 3x\) to express the answer in terms of \(x\).

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Key Concepts

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

Integration of Functions Involving Square Roots

This concept involves integrating functions where the integrand contains a square root expression, often requiring substitution or recognition of standard integral forms. Understanding how to manipulate and simplify the integrand is crucial for solving such integrals.
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Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² + x²), √(a² - x²), or √(x² - a²). By substituting x with a trigonometric function, the integral becomes easier to evaluate using trigonometric identities.
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Standard Integral Formulas

Familiarity with standard integral formulas, such as ∫ dx / √(a² + x²) = ln|x + √(x² + a²)| + C, helps quickly evaluate integrals without lengthy calculations. Recognizing these forms allows for efficient problem-solving.
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Circles in Standard Form Example 1
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.

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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

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

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In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

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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)