Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.8
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999
Verified step by step guidance1
Identify the integral to be evaluated: \(\int_0^1 \frac{dr}{r^{0.999}}\).
Rewrite the integrand using exponent rules: \(\frac{1}{r^{0.999}} = r^{-0.999}\), so the integral becomes \(\int_0^1 r^{-0.999} \, dr\).
Recall the power rule for integration: For \(\int r^n \, dr\), where \(n \neq -1\), the antiderivative is \(\frac{r^{n+1}}{n+1} + C\).
Apply the power rule to the integral: \(\int_0^1 r^{-0.999} \, dr = \left[ \frac{r^{-0.999 + 1}}{-0.999 + 1} \right]_0^1 = \left[ \frac{r^{0.001}}{0.001} \right]_0^1\).
Evaluate the definite integral by substituting the limits: calculate \(\frac{1^{0.001}}{0.001} - \frac{0^{0.001}}{0.001}\), noting the behavior of \(r^{0.001}\) as \(r\) approaches 0.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integrands that are unbounded or have infinite limits. When the integrand approaches infinity at a boundary, the integral is evaluated as a limit to determine convergence or divergence.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Power Rule for Integration
The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1. This rule is essential for integrating functions with variable exponents, allowing direct evaluation of integrals involving powers of the variable.
Recommended video:
04:04Power Rule for Indefinite Integrals
Convergence of Integrals with Singularities
When the integrand has a singularity (like r^(-0.999) near zero), determining if the integral converges requires analyzing the behavior near the singularity. If the integral's limit exists and is finite, the integral converges.
Recommended video:
07:51Choosing a Convergence Test
Related Practice
Textbook Question
Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = c * x * √(25 - x²) over [0, 5]
1
views
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x² dx) / (x² - 1)^(5/2), where x > 1
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
1
views
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).
Textbook Question
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].