Textbook Question
11-14. {Use of Tech} Compute the absolute and relative errors in using c to approximate x.
12. x = √2; c = 1.414
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.4.62
60–69. Completing the square Evaluate the following integrals.
62. ∫ du / (2u² - 12u + 36)
Verified step by step guidance1
Start by examining the quadratic expression in the denominator: \(2u^{2} - 12u + 36\). To simplify the integral, factor out the coefficient of \(u^{2}\) from the quadratic terms.
Rewrite the denominator as \(2(u^{2} - 6u + 18)\) by factoring out the 2. This will make completing the square inside the parentheses easier.
Complete the square for the expression inside the parentheses: \(u^{2} - 6u + 18\). Recall that to complete the square for \(u^{2} - 6u\), add and subtract \((\frac{6}{2})^{2} = 9\) inside the expression.
Rewrite \(u^{2} - 6u + 18\) as \((u - 3)^{2} + (18 - 9)\), which simplifies to \((u - 3)^{2} + 9\). Now the denominator is \(2[(u - 3)^{2} + 9]\).
Substitute \(w = u - 3\) to simplify the integral to the form \(\int \frac{du}{2(w^{2} + 9)}\). Then, factor out constants and use the standard integral formula for \(\int \frac{dx}{x^{2} + a^{2}}\) to proceed with integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Completing the Square
Completing the square is a technique used to rewrite a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This simplifies integration by transforming the denominator into a recognizable form, such as (u - h)² + k, which is easier to integrate using standard formulas.
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Completing the Square to Rewrite the Integrand
Integration of Rational Functions
Integrating rational functions often involves manipulating the integrand into a simpler form. When the denominator is a quadratic expression, rewriting it by completing the square allows the use of standard integral formulas involving arctangent or logarithmic functions, depending on the form of the quadratic.
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6:04Intro to Rational Functions
Standard Integral Formulas Involving Quadratics
Certain integrals involving quadratics have standard results, such as ∫ dx / (x² + a²) = (1/a) arctan(x/a) + C. Recognizing the denominator as a sum of squares after completing the square enables direct application of these formulas, facilitating the evaluation of the integral.
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Related Practice
Textbook Question
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7–64. Integration review Evaluate the following integrals.
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7–84. Evaluate the following integrals.
13. ∫ [1 / (eˣ √(1 – e²ˣ))] dx
Textbook Question
82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
82. ∫ (from -∞ to -1) dx/(x - 1)⁴
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