Textbook Question
76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ
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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 8.9.42
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
42. ∫ (from 3 to 4) 1/(x-3)³ᐟ² dx
Verified step by step guidance1
Identify the type of improper integral: The integrand is \( \frac{1}{(x-3)^{3/2}} \), and the lower limit of integration is \( x=3 \), where the denominator becomes zero, indicating a potential vertical asymptote and an improper integral at the lower limit.
Rewrite the integral as a limit to handle the improper behavior at \( x=3 \): Express the integral as \( \lim_{t \to 3^+} \int_{t}^{4} \frac{1}{(x-3)^{3/2}} \, dx \). This allows us to evaluate the integral over a domain where the function is defined and then take the limit as \( t \) approaches 3 from the right.
Find the antiderivative of the integrand: Use the power rule for integration by rewriting \( (x-3)^{-3/2} \) and integrating with respect to \( x \). Recall that \( \int (x-a)^n \, dx = \frac{(x-a)^{n+1}}{n+1} + C \) for \( n \neq -1 \).
Evaluate the definite integral from \( t \) to 4 using the antiderivative found in the previous step.
Take the limit as \( t \to 3^+ \) of the evaluated integral to determine if the improper integral converges or diverges.
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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 integrals with infinite limits or integrands that become unbounded within the interval. To evaluate them, we use limits to approach the problematic point and determine if the integral converges or diverges.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Behavior of Functions Near Singularities
When the integrand has a singularity (like a division by zero) inside the interval, we analyze the function's behavior near that point. Understanding how the function behaves near the singularity helps decide if the integral converges or diverges.
Recommended video:
5:46Graphs of Exponential Functions
Evaluating Limits for Convergence
To evaluate an improper integral, we replace the problematic bound with a variable and take the limit as it approaches the singularity. If the limit exists and is finite, the integral converges; otherwise, it diverges.
Recommended video:
07:51Choosing a Convergence Test
Related Practice
Textbook Question
Visual proof Let F(x)=∫₀ˣ √(a²−t²) dt. The figure shows that F(x)= area of sector OAB+ area of triangle OBC.
a. Use the figure to prove that
F(x) = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2
b. Conclude that ∫ √(a²−x²) dx = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2 + C.
Textbook Question
{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C
Graph the following functions and find the area under the curve on the given interval.
f(x) = (9 - x²) ⁻², [0, 3/2]
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Textbook Question
64. (Use of Tech) Normal distribution of movie lengths
A study revealed that the lengths of U.S. movies are normally distributed with a mean of 110 minutes and a standard deviation of 22 minutes. This means that the fraction of movies with lengths between a and b minutes (with a < b) is given by the integral:
(1/(22√(2π))) ∫[a to b] e^(-((x-110)/22)²/2) dx.
What percentage of U.S. movies are between 1 hr and 1.5 hr long (60-90 min)?
Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
40. ∫ e^√x dx
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Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
50. ∫ (from 0 to 9) 1/(x - 1)¹ᐟ³ dx
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