Problem 26 - Calculus Single Variable

Question

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{1 / e}^{e} \frac{1}{x \ln ^{2 / 7}(x)} d x\)

Step-by-Step Solution

Answer

The integral converges, and its value is \( \frac{14}{5} \).

Step 1: Determine the Type of Integral

The given integral is \[ \int_{1/e}^{e} \frac{1}{x \ln^{2/7}(x)} \, dx. \] This is an improper integral because the integrand, \( \frac{1}{x \ln^{2/7}(x)} \), becomes undefined when \( \ln(x) = 0 \). This occurs at \( x = 1 \) which is not within the interval \( \left[ \frac{1}{e}, e \right] \). Therefore, the integral does not have an issue within the given limits and we don't need to split or remake the integral.

Step 2: Check Convergence at the Endpoints

The integral does not have any discontinuities at the endpoints \( \frac{1}{e} \) and \( e \). Therefore, we start by simply analyzing the behavior of the integrand over the interval \( \left[ \frac{1}{e}, e \right] \). The function \( \ln(x) \) is positive in this interval, and the denominator \( x \ln^{2/7}(x) \) does not go to zero or infinity within the given limits.

Step 3: Calculate the Antiderivative

To find if the integral converges, compute the antiderivative: Let \( u = \ln(x) \), then \( du = \frac{1}{x} \, dx \). So the integral becomes:\[ \int \frac{1}{x \ln^{2/7}(x)} \, dx = \int u^{-2/7} \, du. \] The antiderivative of \( u^{-2/7} \) is \( \frac{7}{5} u^{5/7} \). Thus the antiderivative is:\[ \frac{7}{5} \ln(x)^{5/7}. \]

Step 4: Evaluate the Definite Integral

Use the antiderivative to evaluate the definite integral from \( 1/e \) to \( e \): \[ \left[ \frac{7}{5} \ln(x)^{5/7} \right]_{1/e}^{e}. \]Compute:\[ \frac{7}{5} (\ln(e)^{5/7} - \ln(1/e)^{5/7}) = \frac{7}{5} (1^{5/7} - (-1)^{5/7}). \]Simplifying gives:\[ \frac{7}{5} (1 + 1) = \frac{14}{5}. \]

Step 5: Verify Convergence

Since the definite integral evaluates to a finite value \( \frac{14}{5} \), the improper integral converges. The result \( \frac{14}{5} \) is the value of the integral on this interval.

Key Concepts

Convergence of Integrals

Convergence of integrals tells us whether an integral will yield a finite value (convergence) or not (divergence). When dealing with improper integrals, the primary concern is the behavior of the integrand at the boundaries of the interval.
Typically, an improper integral requires special attention as it may become undefined or go to infinity at one of these points.
In our example:

• We see that the integrand becomes undefined if \(\ln(x) = 0\), which happens at \(x = 1\), yet this point isn't within our integration bounds of \[ \frac{1}{e}, e \right. \].
• Since there's no discontinuity or infinite limit in the interval \[ \frac{1}{e}, e \right. \], we conclude that the integral behavior is well defined.

Therefore, it is determined to be convergent if the calculated definite integral results in a finite value.

Antiderivative

Finding the antiderivative, or the indefinite integral, is essential for evaluating definite integrals. An antiderivative of a function is another function whose derivative is the original function.
For our function:

• We substituted \(u = \ln(x)\), which simplified the integration task.
• This transformed the integral into a form that was easier to handle mathematically.
• The calculated antiderivative was \( \frac{7}{5} \ln(x)^{5/7}\).

Thus, knowing how to find antiderivatives allows us to handle definite integrals effectively and determines if the convergence is possible.

Change of Variables

The change of variables is a strategic technique used in integration to simplify the process. By altering the variables, complex parts become more manageable. This step can significantly help solve integrals that might initially appear challenging.
In this exercise:

• We implemented the substitution \(u = \ln(x)\), which simplifies the integral, allowing us to use basic power rule techniques.
• The substitution simplifies an otherwise difficult integral into an easily integrable form \( \int u^{-2/7} \, du\).
• Calculations of the antiderivative became straightforward from this point.

Therefore, the change of variables transforms complex integrations into simpler ones, which is invaluable for solving improper integrals.

Definite Integrals

A definite integral gives us the net area under the curve of a function between two specified points, here from \[\frac{1}{e}\right.\] to \[e\right.\]. The finite result of this calculation will indicate convergence of the integral.
In our scenario:

• Once the antiderivative, \( \frac{7}{5} \ln(x)^{5/7}\), was established, it was used to evaluate the limits of integration properly.
• This provides the result \[ \frac{7}{5} (1^{5/7} - (-1)^{5/7}) = \frac{14}{5} \right.\].
• As the result obtained was a finite figure \[ \frac{14}{5} \right.\], it confirms the integral's convergence.

Thus, definite integrals denote the bounded area and affirm whether improper integrals are convergent based on their calculated finite value.