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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 8.6.6
Chapter 8, Problem 8.6.6

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx

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Start by recognizing that the integral is of the form \(\int x (7x + 5)^{3/2} \, dx\), which suggests using substitution to simplify the expression inside the integral.
Let \(u = 7x + 5\). Then, compute the differential \(du = 7 \, dx\), which implies \(dx = \frac{du}{7}\).
Express \(x\) in terms of \(u\) by rearranging the substitution: \(x = \frac{u - 5}{7}\).
Rewrite the integral entirely in terms of \(u\): replace \(x\) with \(\frac{u - 5}{7}\) and \(dx\) with \(\frac{du}{7}\), so the integral becomes \(\int \left( \frac{u - 5}{7} \right) u^{3/2} \cdot \frac{du}{7}\).
Simplify the constants and the integrand, then use the table of integrals to find the integral of the resulting expression in terms of \(u\). After integrating, substitute back \(u = 7x + 5\) 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 by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function, such as (7x + 5)^(3/2).
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Using a Table of Integrals

A table of integrals provides formulas for common integral forms, allowing quick evaluation without performing integration from first principles. Recognizing the integral's form and matching it to an entry in the table can save time and reduce errors. It is important to adjust the integral to fit the table's format, often using substitution.
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Power Functions and Exponents in Integration

Integrating functions with fractional exponents, like (7x + 5)^(3/2), requires understanding how to handle power functions. The power rule for integration extends to fractional powers, where the integral of x^n is (x^(n+1))/(n+1) plus a constant, provided n ≠ -1. Recognizing this helps in applying the correct formula or substitution.
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Related Practice
Textbook Question

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)

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Textbook Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

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Textbook Question

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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

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

Textbook Question

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.

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