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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 8.5.66
Chapter 8, Problem 8.5.66

Use any method to evaluate the integrals in Exercises 55–66.
∫ x² √(1 - x²) dx

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1
Recognize that the integral involves the expression \(\sqrt{1 - x^2}\), which suggests a trigonometric substitution because \(1 - x^2\) resembles the identity \(\sin^2\theta + \cos^2\theta = 1\).
Use the substitution \(x = \sin\theta\), which implies \(dx = \cos\theta \, d\theta\). This transforms the integral into terms of \(\theta\).
Rewrite the integral \(\int x^2 \sqrt{1 - x^2} \, dx\) as \(\int \sin^2\theta \sqrt{1 - \sin^2\theta} \cos\theta \, d\theta\). Since \(\sqrt{1 - \sin^2\theta} = \cos\theta\), the integral becomes \(\int \sin^2\theta \cos\theta \cos\theta \, d\theta = \int \sin^2\theta \cos^2\theta \, d\theta\).
Simplify the integral to \(\int \sin^2\theta \cos^2\theta \, d\theta\). Use trigonometric identities such as the double-angle formulas to express powers of sine and cosine in terms of cosines of multiple angles, which makes the integral easier to evaluate.
After integrating with respect to \(\theta\), substitute back \(\theta = \arcsin x\) 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 involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can often be transformed into a more manageable form. This method is especially useful when the integral contains composite functions like √(1 - x²).
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Substitution With an Extra Variable

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. For integrals containing √(1 - x²), substituting x = sin(θ) leverages the identity 1 - sin²(θ) = cos²(θ), making the integral easier to evaluate.
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Introduction to Trigonometric Functions

Integration of Polynomial Functions

Understanding how to integrate polynomial functions like x² is fundamental. Polynomials are integrated by increasing the exponent by one and dividing by the new exponent. This basic skill is often combined with substitution methods to solve more complex integrals.
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Taylor Polynomials
Related Practice
Textbook Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

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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.

∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))

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Textbook Question

Expand the quotients in Exercises 1–8 by partial fractions.

(5x - 7) / (x² - 3x + 2)

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Textbook Question

Average value

In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is

y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.

Find the average value of y over the interval 0 ≤ t ≤ 2π.

Textbook Question

Evaluate the integrals in Exercises 1–14.

∫ √(1 - 9t²) dt

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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