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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.PE.36
Chapter 8, Problem 8.PE.36

Evaluate the integrals in Exercises 33–36.
∫ [1 / √(9 - x²)] dx

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1
Recognize that the integral \( \int \frac{1}{\sqrt{9 - x^2}} \, dx \) resembles the standard form \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \), where \( a = 3 \).
Recall the formula for this type of integral: \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C \), 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 \( \int \frac{1}{\sqrt{9 - x^2}} \, dx = \arcsin\left(\frac{x}{3}\right) + C \).
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.
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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.
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Derivatives of Other Inverse Trigonometric Functions

Substitution Method in Integration

Substitution is a technique to simplify integrals by changing variables. For integrals like ∫ dx/√(9 - x²), substituting x = 3 sin θ transforms the integral into a simpler trigonometric form, making it easier to evaluate.
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Euler's Method
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