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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 8.4.8
Chapter 8, Problem 8.4.8

Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt

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Recognize that the integral \( \int \sqrt{1 - 9t^2} \, dt \) resembles the form \( \int \sqrt{a^2 - u^2} \, du \), which suggests using a trigonometric substitution to simplify the square root expression.
Set the substitution \( u = 3t \), so that \( u^2 = 9t^2 \). Then, rewrite the integral in terms of \( u \): \( \int \sqrt{1 - u^2} \cdot \frac{du}{3} \). This simplifies the integral to \( \frac{1}{3} \int \sqrt{1 - u^2} \, du \).
Use the trigonometric substitution \( u = \sin \theta \), which implies \( du = \cos \theta \, d\theta \). This transforms the integral into \( \frac{1}{3} \int \sqrt{1 - \sin^2 \theta} \cdot \cos \theta \, d\theta \).
Simplify the square root using the Pythagorean identity: \( \sqrt{1 - \sin^2 \theta} = \cos \theta \). The integral becomes \( \frac{1}{3} \int \cos \theta \cdot \cos \theta \, d\theta = \frac{1}{3} \int \cos^2 \theta \, d\theta \).
Use the power-reduction formula for \( \cos^2 \theta \): \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \). Substitute this into the integral and integrate with respect to \( \theta \). After integrating, substitute back \( \theta = \arcsin u \) and then \( u = 3t \) to express the answer in terms of \( t \).

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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 evaluate integrals involving square roots of quadratic expressions. By substituting a trigonometric function for the variable, the integral simplifies using trigonometric identities. For example, for integrals with √(a² - x²), substituting x = a sin(θ) helps transform the integral into a trigonometric form.
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Introduction to Trigonometric Functions

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions like sine and cosine. Understanding how to integrate these functions, including using identities and standard integral formulas, is essential. This allows the integral to be evaluated in terms of θ before converting back to the original variable.
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Introduction to Trigonometric Functions

Back-Substitution

Once the integral is evaluated in terms of the trigonometric variable, back-substitution replaces the trigonometric expression with the original variable. This step uses the initial substitution relationship and trigonometric identities to express the final answer in terms of the original variable t.
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Substitution With an Extra Variable
Related Practice
Textbook Question

Use any method to evaluate the integrals in Exercises 55–66.

∫ x² √(1 - x²) dx

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

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

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Textbook Question

Evaluate ∫ x³ √(1 - x²) dx using:

c. A trigonometric substitution.

Textbook Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ e^(z + eᶻ) dz