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Magazine Chapter 7: Problem 33
In this problem, we investigate the integral $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$
Short Answer
Expert verified
The integral converges for \( p > 1 \).
Step by step solution
01
Understand the Integral and its Convergence Criterion
The integral given is \( \int_{1}^{\infty} \frac{1}{x^p} \, dx \). This is an improper integral, so its convergence depends on the value of \( p \). We need to explore the relationship between convergence and \( p \).
02
Integrate with a Limit for Improper Integrals
Use a limit to deal with the improper integral: \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^p} \, dx \). This is necessary because the upper limit is infinity.
03
Perform the Antiderivative Calculation
First, determine the antiderivative of \( \frac{1}{x^p} \). For \( p eq 1 \), integrate to get: \( \int \frac{1}{x^p} \, dx = \frac{x^{1-p}}{1-p} + C \).
04
Evaluate the Definite Integral Using the Antiderivative
For the definite integral, substitute the antiderivative into the integral with limits: \( \lim_{b \to \infty} \left[ \frac{x^{1-p}}{1-p} \right]_{1}^{b} \), which simplifies to \( \lim_{b \to \infty} \left( \frac{b^{1-p}}{1-p} - \frac{1^{1-p}}{1-p} \right) \).
05
Analyze Convergence Based on the Value of \( p \)
Examine the limit: For \( p>1 \), \( b^{1-p} \to 0 \) as \( b \to \infty \), leading to a finite result. For \( p \leq 1 \), the integral doesn't converge, as \( b^{1-p} \to \infty \).
06
Conclusion on Convergence
The integral converges if \( p > 1 \) and diverges if \( p \leq 1 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Convergence Criteria
When dealing with improper integrals, like \( \int_{1}^{\infty} \frac{1}{x^p} \, dx \), a vital concept is determining if the integral converges or diverges. Convergence criteria tell us under which conditions the value of the integral approaches a finite limit as the integration upper bound reaches infinity.
Analyzing Convergence
For the given integral, its behavior heavily depends on the exponent \( p \). Here's how it works:
Analyzing Convergence
For the given integral, its behavior heavily depends on the exponent \( p \). Here's how it works:
- If \( p > 1 \), the term \( b^{1-p} \) (where \( b \to \infty \)) approaches 0, leading the integral to converge, or settle on a finite number.
- If \( p \leq 1 \), the term \( b^{1-p} \) does not diminish and heads towards infinity, causing the integral to diverge.
- The integral converges for \( p > 1 \).
- The integral diverges for \( p \leq 1 \).
Antiderivative Calculation Highlights
Finding the antiderivative of \( \frac{1}{x^p} \) prepares us for solving the improper integral. This calculation is foundational because it helps us establish a basic function that can represent a "primitive" form of the original function 鈥� a reverse of differentiation.
Steps to Calculate the Antiderivative
Steps to Calculate the Antiderivative
- Consider the integrand \( \frac{1}{x^p} \), and manipulate it to become suitable for integration rules.
- If \( p eq 1 \), the antiderivative is computed as: \(\int \frac{1}{x^p} \, dx = \frac{x^{1-p}}{1-p} + C \), where \( C \) is the constant of integration.
- When \( p = 1 \), notice the formula becomes undefined. Thus, the integral \( \int \frac{1}{x} \, dx = \ln|x| + C \), since the exploration of \( x^{-1} \) is distinct.
Integral Calculus and Its Role
Integral calculus primarily deals with the concept of integration, which is essentially the idea of summing continuous values over a specified interval. In scenarios of improper integrals, where one or both bounds are infinite or the integrand becomes infinite at some point, careful approaches are required.
The Process in a Nutshell
The Process in a Nutshell
- The first task is to understand the nature of the integral. This involves determining whether it is proper or improper, requiring techniques like limits to properly handle boundaries at infinity.
- Next, compute the antiderivative function of the integrand. This mathematical tool simplifies further evaluations.
- To evaluate \( \int_{1}^{\infty} \frac{1}{x^p} \, dx \), transform it into a limit: \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^p} \, dx \).
- Apply the antiderivative: calculate \( \lim_{b \to \infty} \left[ \frac{x^{1-p}}{1-p} \right]_{1}^{b} \).
- The final step is to substitute and compute limits to check for convergence or divergence.
