7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
19. ∫ 1/√(x² - 81) dx, x > 9

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, a = 9, so substitute x = 9 sec(θ).
2. Differentiate x = 9 sec(θ) to find dx. Using the derivative of sec(θ), dx = 9 sec(θ) tan(θ) dθ.
3. Substitute x = 9 sec(θ) and dx = 9 sec(θ) tan(θ) dθ into the integral. Also, replace √(x² - 81) using the identity sec²(θ) - 1 = tan²(θ), which simplifies √(x² - 81) to 9 tan(θ).
4. Simplify the integral after substitution. The square root √(x² - 81) becomes 9 tan(θ), and dx becomes 9 sec(θ) tan(θ) dθ. The integral simplifies to ∫ (1 / (9 tan(θ))) * (9 sec(θ) tan(θ)) dθ.
5. Evaluate the simplified integral, which reduces to ∫ sec(θ) dθ. The antiderivative of sec(θ) is ln|sec(θ) + tan(θ)| + C. Finally, revert back to the original variable x using the substitution x = 9 sec(θ) and the trigonometric relationships.

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 involve expressions like √(x² - a²), where a is a constant.

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, facilitating the simplification of integrals. For example, if we let x = a sec(θ), we can use this identity to rewrite √(x² - a²) in terms of trigonometric functions.

Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and trigonometric substitution. Understanding these techniques is essential for solving complex integrals, as they provide different approaches to simplify and compute the integral. Mastery of these methods allows students to tackle a wide range of problems in calculus effectively.