Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.6.53

Chapter 7, Problem 7.6.53

Evaluate the integrals in Exercises 53–76.
53. ∫dx/√(9-x²)

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Recognize that the integral \( \int \frac{dx}{\sqrt{9 - x^2}} \) is of the form \( \int \frac{dx}{\sqrt{a^2 - x^2}} \), where \( a = 3 \).

Recall the standard integral formula: \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C \), where \( C \) is the constant of integration.

Apply the formula by substituting \( a = 3 \) into the expression, giving \( \arcsin\left(\frac{x}{3}\right) + C \).

Write the final integral expression as \( \int \frac{dx}{\sqrt{9 - x^2}} = \arcsin\left(\frac{x}{3}\right) + C \).

Remember to include the constant of integration \( C \) since this is an indefinite integral.

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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

Integrals containing expressions like √(a² - x²) often require recognizing standard forms or using trigonometric substitution. These integrals relate to inverse trigonometric functions, which simplify the integration process.

Inverse Trigonometric Functions

Inverse trigonometric functions such as arcsin, arccos, and arctan arise naturally when integrating functions involving square roots of quadratic expressions. For example, ∫dx/√(a² - x²) equals arcsin(x/a) + C.

Trigonometric Substitution

Trigonometric substitution replaces variables with trigonometric expressions to simplify integrals involving radicals. For √(a² - x²), substituting x = a sin θ transforms the integral into a form easier to integrate.

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Related Practice

Textbook Question

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = x³ + 1

Textbook Question

Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.

f(x) = (1 − x)³

Textbook Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

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Evaluate the integrals in Exercises 33–54.

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