Question

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.\(\int_{1}^{\infty} \frac{d x}{x^{2}+4 x} ;\) compare with \(\int_{1}^{\infty} \frac{d x}{x^{2}}\)

Short answer

The integral \( \int_{1}^{\infty} \frac{d x}{x^{2}+4 x} \) converges to \( \frac{1}{4} \ln(5) \).

Step-by-step solution

Simplify the Given Function

Start by simplifying the given function \( \frac{1}{x^2 + 4x} \). Rewrite it as \( \frac{1}{x(x+4)} \).

Choose a Comparison Function

We will compare with the integral \( \int_{1}^{\infty} \frac{d x}{x^2} \). Notice that \( \frac{1}{x^2+4x} \leq \frac{1}{x^2} \) for \( x \geq 1 \).

Analyze the Comparison Function

Evaluate \( \int_{1}^{\infty} \frac{d x}{x^2} \). This is an improper integral of the form \( x^{-p} \) with \( p=2 > 1 \), which is known to converge to \(-\frac{1}{x}\bigg|_1^\infty = 1 \).

Determine Convergence of the Integral

Since \( \int_{1}^{\infty} \frac{d x}{x^2} \) converges and \( \frac{1}{x(x+4)} \leq \frac{1}{x^2} \), by the comparison test, \( \int_{1}^{\infty} \frac{d x}{x^2+4x} \) also converges.

Calculate the Convergent Value

To find the exact value, perform partial fraction decomposition: \( \frac{1}{x(x+4)} = \frac{A}{x} + \frac{B}{x+4} \). Solve for \( A \) and \( B \), giving you \( A = \frac{1}{4} \) and \( B = -\frac{1}{4} \).

Evaluate the Integral

Integrate the function \( \frac{1}{4x} - \frac{1}{4(x+4)} \). Evaluate the integrals: \( \int \frac{1}{4x} \, dx = \frac{1}{4} \ln|x| \) and \( \int \frac{1}{4(x+4)} \, dx = \frac{1}{4} \ln|x+4| \).

Compute the Limits

Compute the limit as \( x \to \infty \) and \( x = 1 \): \( \left[ \frac{1}{4} \ln\left| \frac{x}{x+4} \right| \right]_1^\infty \). As \( x \to \infty \), \( \ln\left|\frac{x}{x+4}\right| \to 0 \). At \( x = 1 \), \( \ln\left|\frac{1}{5}\right| = -\ln(5) \).

Final Convergent Value

Combine the limits to get the convergent value: \( 0 - (-\frac{1}{4} \ln(5)) = \frac{1}{4} \ln(5) \). Thus, the integral converges to \( \frac{1}{4} \ln(5) \).

Key concepts

Improper Integrals

Improper integrals are a critical topic in calculus, especially when dealing with limits of functions over unbounded intervals or with unbounded functions. These integrals occur when the region of integration involves an endpoint of infinity (like from 1 to infinity) or when the integrand takes on infinite values at some points within the region of integration. For students, understanding the concept of improper integrals is essential for tackling a wide range of problems.

• Infinite Limits of Integration: When the upper (or lower) limit of the integral is infinity.
• Discontinuity within Bounds: When the integrand has a vertical asymptote, or point where the function is undefined, within the interval of integration.

The integral we tackled is of the type where the upper limit goes to infinity, requiring us to assess convergence by evaluating the limit as the variable approaches infinity. This kind of integral is often handled with comparison tests to determine if it converges or diverges.

Comparison Test

The comparison test is a powerful tool in determining the convergence or divergence of improper integrals. It involves comparing an integral of interest to a simpler, known integral. By choosing a suitable comparison, one can infer properties about the original integral.

• Selection of Comparison Function: The function chosen for comparison must closely approximate the behavior of the integrand. Ideally, it should be a function whose integral's convergence is already known.
• Establishing Inequality: If \( f(x) \leq g(x) \) for \( x \) in \( [a, \infty) \,\) and \( \int_a^{\infty} g(x) \; dx \) converges, then \( \int_a^{\infty} f(x) \; dx \) also converges.

In the original problem, we compared \( \int_{1}^{\infty} \frac{dx}{x^2+4x} \) with \( \int_{1}^{\infty} \frac{dx}{x^2}\). Since the latter is known to converge, by the comparison test, the former also converges.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify complex rational expressions, making them easier to integrate or differentiate. This method involves expressing a rational function as a sum of simpler fractions, which can then be integrated individually.

• Breaking Down the Fraction: For \( \frac{1}{x(x+4)} \, \) express it as \( \frac{A}{x} + \frac{B}{x+4} \).
• Solving for Constants: Multiply both sides by the denominator to eliminate the fractions and equate coefficients to solve for \( A \) and \( B \).

In this exercise, decomposing the function into separate parts simplified the integration process. We found the values of \( A = \frac{1}{4} \) and \( B = -\frac{1}{4} \), allowing us to calculate each integral part.

Improper Integrals Convergence

Understanding the convergence of improper integrals is crucial when dealing with unbounded or discontinuous functions. Convergence indicates that as we integrate, the accumulated value approaches a finite limit, whereas divergence means it tends to infinity.

• Using Known Results: Certain forms of integrals like \( \int_1^{\infty} \frac{1}{x^p} \; dx \, \) converge if \( p > 1 \), providing a useful reference for similar integrals.
• Comparative Analysis: When unsure about an integral's convergence, compare it to one with a known behavior. If the simpler integral converges and bounds the original, so does the original.

In the exercise, confirming the convergence of \( \int_{1}^{\infty} \frac{1}{x^2} \) with \( p = 2 \) helped validate that our integral of interest is convergent.

Convergent Values Calculation

After establishing convergence, the next step is to calculate the exact convergent value of the integral, if possible. This is where integration techniques like partial fractions come into play.

• Integrating Decomposed Fractions: Calculate each part of the decomposed expression separately. For instance, \( \int \frac{1}{4x} \; dx \) gives \( \frac{1}{4} \ln|x| \).
• Evaluating Limits: Determine the definite integral by calculating the limits of the indefinite integral as the variable approaches the bounds.
• Combining Results: Sum or subtract the evaluated limits to find the final convergent value of the integral.

For our problem, through careful decomposition and integration, we concluded that the integral converges to \( \frac{1}{4} \ln(5) \, \) demonstrating the application of these steps effectively.