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

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1. Recognize that the integral \( \int \frac{dx}{\sqrt{9 - x^2}} \) is of the form \( \int \frac{dx}{\sqrt{a^2 - x^2}} \), where \( a = 3 \).
2. 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.
3. Apply the formula by substituting \( a = 3 \) into the expression, giving \( \arcsin\left(\frac{x}{3}\right) + C \).
4. Write the final integral expression as \( \int \frac{dx}{\sqrt{9 - x^2}} = \arcsin\left(\frac{x}{3}\right) + C \).
5. Remember to include the constant of integration \( C \) since this is an indefinite integral.

Key Concepts

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.