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

검증된 단계별 안내

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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주요 개념

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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.

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.

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