Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
c. 2π ≤ x ≤ 3π.
44
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.1.54c
Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
c. u = arctan x
What is the value of the integral?
Verified step by step guidance1
Start with the integral \( \int ((x^{2} - 1)(x + 1))^{-\frac{2}{3}} \, dx \). First, simplify the expression inside the integral if possible. Notice that \( (x^{2} - 1)(x + 1) = (x - 1)(x + 1)(x + 1) = (x - 1)(x + 1)^{2} \). So the integral becomes \( \int \left( (x - 1)(x + 1)^{2} \right)^{-\frac{2}{3}} \, dx \).
Apply the substitution \( u = \arctan x \). Then, differentiate both sides to find \( du \) in terms of \( dx \): \( du = \frac{1}{1 + x^{2}} \, dx \), which implies \( dx = (1 + x^{2}) \, du \).
Express the integral entirely in terms of \( u \). Since \( x = \tan u \), rewrite all parts of the integrand: \( x - 1 = \tan u - 1 \), \( x + 1 = \tan u + 1 \), and \( dx = (1 + \tan^{2} u) \, du = \sec^{2} u \, du \). Substitute these into the integral.
Rewrite the integral as \( \int \left( (\tan u - 1)(\tan u + 1)^{2} \right)^{-\frac{2}{3}} \sec^{2} u \, du \). Simplify the expression inside the parentheses if possible, and then simplify the powers and factors to get the integral in terms of \( u \) only.
Once the integral is expressed in terms of \( u \), proceed to integrate with respect to \( u \). After finding the antiderivative, substitute back \( u = \arctan x \) to express the answer in terms of \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution is a method to simplify integrals by changing variables. It involves choosing a new variable u = g(x) to rewrite the integral in terms of u, making it easier to evaluate. The derivative du/dx replaces dx, and the integral limits or integrand are adjusted accordingly.
Recommended video:
04:27
Substitution With an Extra Variable
Trigonometric Substitution
Trigonometric substitution replaces algebraic expressions with trigonometric functions to simplify integrals involving roots or powers. For example, using u = arctan(x) transforms expressions involving x into functions of u, leveraging identities like 1 + tan²u = sec²u to simplify the integrand.
Recommended video:
6:04Introduction to Trigonometric Functions
Evaluating Integrals with Inverse Trigonometric Functions
When the substitution involves inverse trigonometric functions, the integral is expressed in terms of u = arctan(x). This requires rewriting x and dx in terms of u and using trigonometric identities to simplify the integrand before integrating, often resulting in expressions involving trigonometric or algebraic functions of u.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
89. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / x², y = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.
19
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Textbook Question
The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from 0 to π of sin(t) dth
16
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Textbook Question
The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from -2 to 0 of (x² - 1) dx
23
views
Textbook Question
90. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / √x, y = 0, x = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.
25
views
Textbook Question
The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from 1 to 2 of 1 / s² ds
16
views