Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
13. ∫ (from 0 to ∞) cos x 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 8.6.22
7–84. Evaluate the following integrals.
22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b
Verified step by step guidance1
Recognize that the integral involves a rational function with two distinct linear factors in the denominator, (x - a) and (x - b). This suggests using partial fraction decomposition to simplify the integrand.
Express the integrand as a sum of partial fractions: \( \frac{1}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b} \), where A and B are constants to be determined.
Multiply through by \((x - a)(x - b)\) to eliminate the denominators, resulting in \( 1 = A(x - b) + B(x - a) \). Expand and collect terms to solve for A and B.
Substitute the values of A and B back into the partial fraction decomposition, rewriting the integrand as \( \frac{A}{x - a} + \frac{B}{x - b} \). Then, split the integral into two separate integrals: \( \int \frac{A}{x - a} dx + \int \frac{B}{x - b} dx \).
Evaluate each integral using the formula \( \int \frac{1}{x - c} dx = \ln|x - c| + C \), where C is the constant of integration. Combine the results to obtain the final expression for the integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration
Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding integration techniques, such as substitution and partial fractions, is essential for evaluating integrals.
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Integration by Parts for Definite Integrals
Partial Fraction Decomposition
Partial fraction decomposition is a method used to break down a complex rational function into simpler fractions that are easier to integrate. This technique is particularly useful when dealing with integrals of rational functions, as it allows for the integration of each simpler fraction separately. In the given integral, the expression 1 / ((x - a)(x - b)) can be decomposed into simpler fractions to facilitate integration.
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Definite vs. Indefinite Integrals
Definite integrals calculate the area under a curve between two specific limits, while indefinite integrals represent a family of functions and include a constant of integration. The integral in the question is an indefinite integral, meaning it will yield a general antiderivative of the function. Understanding the distinction between these two types of integrals is crucial for correctly interpreting the results of integration.
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Related Practice
Textbook Question
7–64. Integration review Evaluate the following integrals.
51. ∫ from -1 to 0 of x / (x² + 2x + 2) dx
Textbook Question
7–84. Evaluate the following integrals.
82. ∫ 1/(1 + tanx) dx
Textbook Question
48. Integral of sec³x Use integration by parts to show that:
∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx
Textbook Question
95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2
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Textbook Question
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt