Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x arctan(x) dx
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.1.10
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₁² (8 dx / (x² - 2x + 2))
Verified step by step guidance1
Start by examining the integral \( \int_1^2 \frac{8}{x^2 - 2x + 2} \, dx \). Notice that the denominator is a quadratic expression that can be rewritten to complete the square.
Rewrite the quadratic in the denominator by completing the square: \( x^2 - 2x + 2 = (x - 1)^2 + 1 \). This form is easier to work with for integration.
Make the substitution \( u = x - 1 \), which implies \( du = dx \). Change the limits of integration accordingly: when \( x = 1 \), \( u = 0 \); when \( x = 2 \), \( u = 1 \). The integral becomes \( \int_0^1 \frac{8}{u^2 + 1} \, du \).
Recognize that the integral \( \int \frac{1}{u^2 + 1} \, du \) corresponds to the inverse tangent function \( \arctan(u) \). Use this fact to express the integral in terms of \( \arctan(u) \).
Integrate using the formula \( \int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \arctan\left( \frac{u}{a} \right) + C \). Since \( a = 1 \), the integral simplifies to \( 8 \arctan(u) \) evaluated from \( u = 0 \) to \( u = 1 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Rational Functions
This involves integrating functions expressed as ratios of polynomials. Techniques often include algebraic manipulation such as completing the square or partial fraction decomposition to simplify the integrand before integration.
Recommended video:
6:04
Intro to Rational Functions
Completing the Square
Completing the square rewrites a quadratic expression into the form (x - h)² + k, which can simplify the integral, especially when the denominator is a quadratic. This form often allows the use of standard integral formulas involving inverse trigonometric functions.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Integration Using Trigonometric Substitution
When the integrand contains expressions like a² + x² or (x - h)² + k, trigonometric substitution can transform the integral into a simpler trigonometric integral. This method leverages identities such as 1 + tan²θ = sec²θ to facilitate integration.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
Textbook Question
Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.
1
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx