7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
33. ∫ √(x² - 9)/x dx, x > 3

Verified step by step guidance

1. Recognize that the integral involves a square root of the form √(x² - a²), which suggests using the trigonometric substitution x = a sec(θ). Here, let x = 3 sec(θ), where a = 3.
2. Substitute x = 3 sec(θ) into the integral. Recall that dx = 3 sec(θ) tan(θ) dθ and √(x² - 9) becomes √((3 sec(θ))² - 9) = √(9 sec²(θ) - 9) = √(9(tan²(θ))) = 3 tan(θ).
3. Rewrite the integral using the substitution: ∫ √(x² - 9)/x dx becomes ∫ (3 tan(θ))/(3 sec(θ)) * 3 sec(θ) tan(θ) dθ. Simplify the expression to ∫ tan²(θ) dθ.
4. Use the trigonometric identity tan²(θ) = sec²(θ) - 1 to rewrite the integral: ∫ tan²(θ) dθ = ∫ (sec²(θ) - 1) dθ. Split the integral into two parts: ∫ sec²(θ) dθ - ∫ 1 dθ.
5. Evaluate the two integrals separately: ∫ sec²(θ) dθ = tan(θ) and ∫ 1 dθ = θ. After integration, substitute back θ using the original substitution x = 3 sec(θ), where sec(θ) = x/3, and tan(θ) = √(x² - 9)/3.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as sine or cosine, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(x² - a²), allowing for easier integration.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is fundamental in trigonometric substitution as it allows us to express one trigonometric function in terms of another. When substituting variables, this identity helps to simplify the resulting expressions, making it easier to evaluate the integral.

Integration Techniques

Integration techniques encompass various methods used to solve integrals, including substitution, integration by parts, and partial fractions. Understanding these techniques is crucial for evaluating complex integrals, such as those involving trigonometric substitutions. Mastery of these methods enables students to tackle a wide range of problems in calculus effectively.