Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ z(ln z)² dz
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 8.8.72
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / x^(1/5)) dx
Verified step by step guidance1
Identify the type of improper integral: Since the upper limit of integration is infinity, this is an improper integral of the form \(\int_1^{\infty} \frac{1}{x^{1/5}} \, dx\).
Rewrite the integrand using exponent notation: \(\frac{1}{x^{1/5}}\) can be written as \(x^{-1/5}\).
Set up the integral with a limit to handle the improper integral: \(\lim_{t \to \infty} \int_1^t x^{-1/5} \, dx\).
Find the antiderivative of \(x^{-1/5}\): Use the power rule for integration, which states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\). Here, \(n = -\frac{1}{5}\), so the antiderivative is \(\frac{x^{4/5}}{4/5} = \frac{5}{4} x^{4/5}\).
Evaluate the definite integral from 1 to \(t\) and then take the limit as \(t\) approaches infinity: Calculate \(\lim_{t \to \infty} \left[ \frac{5}{4} t^{4/5} - \frac{5}{4} \cdot 1^{4/5} \right]\) to determine if the integral converges or diverges.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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 infinite interval or integrands with infinite discontinuities. To evaluate them, we take limits of definite integrals as the interval approaches infinity or the point of discontinuity. Determining convergence means checking if this limit exists and is finite.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
p-integral Test
The p-integral test helps determine convergence of integrals of the form ∫₁^∞ 1/x^p dx. If p > 1, the integral converges; if p ≤ 1, it diverges. This test is essential for quickly assessing the behavior of power function integrals over infinite intervals.
Recommended video:
07:25Integral Test
Evaluating Limits of Integrals
To evaluate an improper integral, express it as a limit of a definite integral with an upper bound t approaching infinity. Compute the antiderivative, substitute the limits, and then take the limit as t → ∞. This process confirms convergence and provides the integral's value if it converges.
Recommended video:
05:50One-Sided Limits
Related Practice
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.
∫ (dθ / √(2θ - θ²))
Textbook Question
Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.
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.
∫ x √(x² - 4) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) dx
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.
∫ (dt / t√(3 + t²)
1
views