Problem 33 - Calculus for Biology and Medicine

Question

In this problem, we investigate the integral $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$

Step-by-Step Solution

Answer

The integral converges for \( p > 1 \).

Step 1: 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 \( p \).

Step 2: 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.

Step 3: Perform the Antiderivative Calculation

First, determine the antiderivative of \( \frac{1}{x^p} \). For \( p eq 1 \), integrate to get: \( \int \frac{1}{x^p} \, dx = \frac{x^{1-p}}{1-p} + C \).

Step 4: 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 \( \lim_{b \to \infty} \left( \frac{b^{1-p}}{1-p} - \frac{1^{1-p}}{1-p} \right) \).

Step 5: 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 \( b^{1-p} \to \infty \).

Step 6: Conclusion on Convergence

The integral converges if \( p > 1 \) and diverges if \( p \leq 1 \).

Key Concepts

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:

• 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.

By examining the limit as part of the integration process, we are given a clear rule:

• The integral converges for \( p > 1 \).
• The integral diverges for \( p \leq 1 \).

These criteria are essential in many fields that require calculations involving infinite processes, providing a foundation for understanding when these processes yield useful numerical results.

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 – a reverse of differentiation.
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.

Special Case

• 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.

The antiderivative acts like a stepping stone, enabling calculations that help to resolve whether an improper integral will reach a finite result, tying directly back to our convergence discussions.

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 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.

Evaluating Improper Integrals

• 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.

Integral calculus brings together these calculations, enabling a deeper understanding of accumulative processes and applications across science and engineering. Knowing how to handle improper integrals is crucial for anyone venturing into higher mathematics or its applications.