Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.4.30
Chapter 8, Problem 8.4.30

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1

Verified step by step guidance
1
Identify the integral to solve: \(\displaystyle \int \frac{x^{2}}{(x^{2} - 1)^{5/2}} \, dx\), with the condition \(x > 1\).
Recognize that the integrand contains the expression \(x^{2} - 1\) under a fractional power, suggesting a trigonometric substitution related to \(\sec \theta\), since \(\sec^{2} \theta - 1 = \tan^{2} \theta\).
Make the substitution \(x = \sec \theta\), which implies \(dx = \sec \theta \tan \theta \, d\theta\). Also, rewrite the denominator: \(x^{2} - 1 = \sec^{2} \theta - 1 = \tan^{2} \theta\).
Rewrite the integral entirely in terms of \(\theta\): replace \(x^{2}\) with \(\sec^{2} \theta\), \((x^{2} - 1)^{5/2}\) with \((\tan^{2} \theta)^{5/2} = \tan^{5} \theta\), and \(dx\) with \(\sec \theta \tan \theta \, d\theta\). Simplify the resulting expression before integrating.
After simplification, integrate the resulting trigonometric expression with respect to \(\theta\). Once integrated, substitute back \(\theta = \sec^{-1} x\) to express the answer in terms of \(x\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
13m
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 to simplify integrals involving expressions like √(x² - a²), √(a² - x²), or √(x² + a²). By substituting x with a trigonometric function, the integral transforms into a trigonometric integral that is easier to evaluate. For example, for √(x² - 1), substituting x = sec(θ) is common.
Recommended video:
6:04
Introduction to Trigonometric Functions

Integration of Rational Functions

Integrals involving rational functions, where the integrand is a ratio of polynomials, often require algebraic manipulation such as polynomial division or rewriting the integrand to simplify the expression before integrating. Recognizing when to apply these techniques helps in breaking down complex integrals.
Recommended video:
6:04
Intro to Rational Functions

Differential and Substitution Method

The substitution method involves changing variables to simplify the integral. After choosing an appropriate substitution (like x = sec(θ)), the differential dx is expressed in terms of dθ, allowing the integral to be rewritten in a simpler form. Correctly handling the differential is crucial for accurate integration.
Recommended video:
07:33
Euler's Method
Related Practice
Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ dr / r^0.999

1
views
Textbook Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ √(x - 2) / √(x - 1) dx

Textbook Question

Use any method to evaluate the integrals in Exercises 55–66.

∫ (x + 2) / (x³ - 2x² - 3x) dx

Textbook Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ (ln x)³/x dx

1
views
Textbook Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x² sin(x³) dx

Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀^∞ dx / [(x + 1)(x² + 1)]

1
views