Textbook Question
114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).
c. As a increases, L(a) increases as what power of a?
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.R.12
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
12. ∫ (8x + 5)/(2x² + 3x + 1) dx
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Identify the integral to solve: \(\int \frac{8x + 5}{2x^{2} + 3x + 1} \, dx\).
Factor the denominator if possible. The quadratic \(2x^{2} + 3x + 1\) factors as \((2x + 1)(x + 1)\).
Express the integrand as a sum of partial fractions: \(\frac{8x + 5}{(2x + 1)(x + 1)} = \frac{A}{2x + 1} + \frac{B}{x + 1}\), where \(A\) and \(B\) are constants to be determined.
Multiply both sides by the denominator \((2x + 1)(x + 1)\) to get: \(8x + 5 = A(x + 1) + B(2x + 1)\). Then, expand and collect like terms to form an equation in \(x\).
Solve the system of equations for \(A\) and \(B\) by equating coefficients of \(x\) and the constant terms. Once \(A\) and \(B\) are found, rewrite the integral as the sum of two simpler integrals and integrate each using the natural logarithm rule: \(\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln|ax + b| + C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down a rational function into simpler fractions that are easier to integrate. It is especially useful when the denominator can be factored into linear or quadratic terms. This method transforms the integral into a sum of simpler integrals.
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Partial Fraction Decomposition: Distinct Linear Factors
Integration of Rational Functions
Integrating rational functions involves expressing the integrand as a ratio of polynomials and applying algebraic techniques like substitution or partial fractions. Recognizing the form of the integrand helps determine the appropriate method to simplify and integrate the expression.
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6:04Intro to Rational Functions
Substitution Method
The substitution method simplifies integration by changing variables to transform the integral into a more manageable form. It is often used when the integrand contains a function and its derivative, allowing the integral to be rewritten in terms of a single variable.
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07:33Euler's Method
Related Practice
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx
Textbook Question
92. Integral with a parameter For what values of p does the integral
∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?
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Textbook Question
7–64. Integration review Evaluate the following integrals.
12. ∫ from -5 to 0 of dx / √(4 - x)
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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Textbook Question
108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].