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 8.4.33
Chapter 8, Problem 8.4.33

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.
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).
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.
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.
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.

Verified video answer for a similar problem:

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

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 contain expressions like √(x² - a²), allowing for easier integration.
Recommended video:
Guided course
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. When substituting variables, this identity helps to simplify the resulting expressions, making it easier to evaluate the integral.
Recommended video:
7:17
Verifying Trig Equations as Identities

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

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

51. ∫ x²/√(4 + x²) dx

Textbook Question

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

26. ∫ t³ sin(t) dt

1
views
Textbook Question

Evaluate the following integrals.

∫ e³ˣ/(eˣ - 1) dx

Textbook Question

7–84. Evaluate the following integrals.

47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

1
views
Textbook Question

29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

33. ∫(0 to π) sin x cos(3x) dx = 0

1
views
Textbook Question

7–84. Evaluate the following integrals.

33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0