Question

For what values of \(p>0\) is \(\int_{0}^{\infty} \frac{d x}{x^{p}+x^{-p}}<\infty ?\)

Short answer

$$ \int_0^\infty \frac{dx}{x^p + x^{-p}} $$ Answer: The integral converges for \(p>1\).

Step-by-step solution

Define where to analyze for convergence

First, we will examine the behavior of the integrand at \(x\to0\) (the lower limit) and \(x\to\infty\) (the upper limit) separately.

Convergence near 0

To study the convergence of the integral near \(x=0\), we will compare our integrand with a simpler function. As \(x\to0\), we have: $$ \frac{1}{x^{p}+x^{-p}}\sim\frac{1}{x^{-p}} $$ Let's now consider the integral $$ \int_{0}^{1} \frac{d x}{x^{-p}}= \int_{0}^{1} x^p dx $$ By using the power rule for integration, integrating with respect to \(x\) will yield: $$ \int_{0}^{1} x^p dx = \left[\frac{x^{p+1}}{p+1}\right]_0^1 = \frac{1}{p+1} $$ This integral converges for all values of \(p>-1\). Thus, for our original integral to converge near \(x=0\), we need \(p>-1\).

Convergence at infinity

To study the convergence of the integral at \(x\to\infty\), we will again compare our integrand with a simpler function. As \(x\to\infty\), we have: $$ \frac{1}{x^{p}+x^{-p}}\sim\frac{1}{x^{p}} $$ Let's consider the integral $$ \int_{1}^{\infty} \frac{d x}{x^{p}}= \int_{1}^{\infty} x^{-p} dx $$ By using the power rule for integration, integrating with respect to \(x\) will yield: $$ \int_{1}^{\infty} x^{-p} dx = \left[\frac{x^{-p+1}}{-p+1}\right]_1^\infty = \frac{1}{p-1} $$ This integral converges for values of \(p>1\). Thus, for our original integral to converge at \(x\to\infty\), we need \(p>1\).

Combining the results

We found that the integral converges near 0 for \(p>-1\) and converges at infinity for \(p>1\). To ensure convergence for the whole range of the integral (\([0,\infty]\)), both conditions must be satisfied simultaneously. Therefore, the values of \(p\) for which the integral converges are: $$ p>1 $$

Key concepts

Convergence

Convergence in mathematical terms refers to the idea of whether a sequence or function reaches a finite limit as it approaches infinity or some singular point. In the context of improper integrals, like the one in our problem, convergence means determining whether the integral reaches a finite value over an infinite interval.

For this exercise, the integral is examined for two limits: near zero and towards infinity. These are the areas that often pose challenges in terms of convergence. Before we apply any integration techniques, we need to ensure the integral doesn't blow up to infinity in these regions.

The analysis method involves comparing the given integrand with simpler functions that have known convergence properties. This comparison helps identify convergence at each of the troublesome limits.

• Convergence near zero requires that the function behaves in a controlled and finite manner as it approaches zero.
• Convergence at infinity ensures that as the input becomes very large, the function's output doesn't diverge to infinity.

For this particular integral, convergence near zero demands that the power of the variable is greater than a specific benchmark, while converging as it approaches infinity requires a different threshold.

Integration Techniques

Integration techniques are the methods used to solve integrals, and they cover a wide range of approaches. In this exercise, these techniques help handle improper integrals by simplifying the integrand as much as possible. A key technique shown here is the use of substitution with known simpler functions.

The integral \[\int_{0}^{\infty} \frac{d x}{x^{p}+x^{-p}}\]can be tricky at first glance. Therefore, it's broken down into more manageable parts. For example, the solution compares the integrand with \(\frac{1}{x^{-p}}\) near zero and \(\frac{1}{x^p}\) towards infinity. These simpler forms are easier to integrate while revealing convergence behavior.

Here are some common techniques used in integration:

• Substitution: Transforming the variable of integration to simplify the integral.
• By parts: Useful for the integral of products of functions.
• Partial fractions: Breaking down complex fractions into simpler ones.

In this example, the analysis method involving comparison effectively helps to reveal the region's behavior of the function by changing complex expressions into fundamental ones that match known integrals.

Power Rule

The power rule is a straightforward integration technique for expressions in the form \(x^n\). It's a fundamental tool in calculus used to integrate polynomials. Applying this rule helps find the antiderivative of \(x^n\) with ease.

For the power rule to be applied, the integral typically takes the form \(\int x^n \, dx\). The solution \(\frac{x^{n+1}}{n+1}\) results from integrating, provided \(neq-1\). This rule is particularly useful when dealing with simple forms of functions, as seen in our exercise.

When examining integrals like those in the solution, the power rule is applied to quickly evaluate whether the integral converges based on the limits:

• Near zero: We use the power rule on the integral \(\int_{0}^{1} x^p \, dx\), which integrates to \(\frac{x^{p+1}}{p+1}\).
• At infinity: The integral \(\int_{1}^{\infty} x^{-p} \, dx\) relies on the power rule to assess behavior, yielding \(\frac{x^{-p+1}}{-p+1}\).

The role of the power rule in this solution demonstrates how a fundamental calculus technique anchors more complex analyses of unconverged series.