Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
33. ∫ √(x² - 9)/x dx, x > 3
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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 8.9.44
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
44. ∫ (from 0 to ln 3) eʸ/(eʸ-1)⁷ᐟ³ dy
Verified step by step guidance1
Identify the integral to evaluate: \(\int_0^{\ln 3} \frac{e^y}{(e^y - 1)^{7/3}} \, dy\).
Check the behavior of the integrand at the limits of integration, especially near \(y=0\) and \(y=\ln 3\), to determine if the integral is improper. Note that at \(y=0\), \(e^y - 1 = 0\), which may cause a singularity.
To analyze the singularity at \(y=0\), perform a substitution or use a limit approach. For example, let \(x = e^y - 1\), so when \(y \to 0\), \(x \to 0\). Express the integrand in terms of \(x\) and examine the limit as \(x \to 0\).
Rewrite the integral using the substitution \(x = e^y - 1\), which implies \(dx = e^y dy = (x + 1) dy\), so \(dy = \frac{dx}{x + 1}\). Substitute into the integral and adjust the limits accordingly.
Evaluate the integral or determine convergence by analyzing the behavior of the integrand near the singularity and then proceed with integration techniques such as substitution or integration by parts if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integration over an unbounded interval or integrands with unbounded behavior within the interval. To evaluate them, one often takes limits approaching the problematic points to determine convergence or divergence.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Behavior of the Integrand Near Singularities
Analyzing the integrand's behavior near points where it may become infinite or undefined is crucial. For example, if the denominator approaches zero, the integrand may have a singularity, affecting convergence of the integral.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Substitution and Simplification Techniques
Using substitution can simplify the integral, especially when the integrand involves composite functions like exponentials. Simplifying the expression helps in identifying limits and evaluating the integral more easily.
Recommended video:
04:27Substitution With an Extra Variable
Related Practice
Textbook Question
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
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Textbook Question
41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.
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Textbook Question
7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)
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Textbook Question
65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
72. The region bounded by f(x) = (x + 1)^(-3/2) and the x-axis on the interval (-1, 1] is revolved about the line y = -1.
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Textbook Question
7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx
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