Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx
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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 8.4.65
60–69. Completing the square Evaluate the following integrals.
65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)
Verified step by step guidance1
Start by examining the quadratic expression in the denominator: \(8x^{2} - 8x + 11\). To simplify the integral, we want to complete the square for this quadratic.
Factor out the coefficient of \(x^{2}\) from the first two terms: \(8(x^{2} - x) + 11\).
Complete the square inside the parentheses: take half of the coefficient of \(x\), which is \(-1\), divide by 2 to get \(-\frac{1}{2}\), then square it to get \(\left(-\frac{1}{2}\right)^{2} = \frac{1}{4}\). Add and subtract this inside the parentheses:
\(8\left(x^{2} - x + \frac{1}{4} - \frac{1}{4}\right) + 11 = 8\left(\left(x - \frac{1}{2}\right)^{2} - \frac{1}{4}\right) + 11\).
Distribute the 8 and simplify the constant terms: \(8\left(x - \frac{1}{2}\right)^{2} - 8 \times \frac{1}{4} + 11 = 8\left(x - \frac{1}{2}\right)^{2} - 2 + 11 = 8\left(x - \frac{1}{2}\right)^{2} + 9\).
Rewrite the integral using this completed square form: \(\int_{\frac{1}{2}}^{\frac{\sqrt{2} + 3}{2\sqrt{2}}} \frac{dx}{8\left(x - \frac{1}{2}\right)^{2} + 9}\). Next, factor out the 9 to express the denominator in a form suitable for an arctangent substitution.
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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 quadratic expressions in the form ax² + bx + c as a perfect square plus or minus a constant. This simplifies integration by transforming the denominator into a form that matches standard integral formulas, especially those involving inverse trigonometric functions.
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Completing the Square to Rewrite the Integrand
Integration of Rational Functions
Integrating rational functions often involves algebraic manipulation such as factoring or completing the square. Recognizing the form of the denominator helps in applying appropriate substitution or standard integral results, enabling the evaluation of integrals that might otherwise be complex.
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6:04Intro to Rational Functions
Definite Integrals and Limits of Integration
Definite integrals calculate the net area under a curve between two specific points. Understanding how to apply the limits after finding the antiderivative is crucial, as it involves substituting the upper and lower bounds correctly to find the exact value of the integral.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].
Textbook Question
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx
Textbook Question
4. Is a reduction formula an analytical method or a numerical method? Explain.
Textbook Question
76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.
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Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
29. ∫ e⁻ˣ sin(4x) dx