Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.4.19
Chapter 8, Problem 8.4.19

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).
Differentiate x = 9 sec(θ) to find dx. Using the derivative of sec(θ), dx = 9 sec(θ) tan(θ) dθ.
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).
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.
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
6:04
Introduction to Trigonometric Functions

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.
Recommended video:
7:17
Verifying Trig Equations as Identities

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.
Recommended video:
06:18
Integration by Parts for Definite Integrals
Related Practice
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

19. ∫[0 to π/3] sin⁵x cos⁻²x dx

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

37. ∫ dx / √(x² + 10x), x >

Textbook Question

92–98. Evaluate the following integrals.

97. ∫ tan⁻¹(∛x) dx

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

42. ∫ 1/(x²√(9x² - 1)) dx, x > 1/3

Textbook Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

36. ∫ from 0 to ln2 x eˣ dx

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

50. ∫ csc¹⁰x cot³x dx