Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. $$ \int_{1}^{2} \frac{d x}{x \ln x} $$

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Analyze the Integrand and Determine Impropriety

The improper integral \( \int_{1}^{2} \frac{d x}{x \ln x} \) is improper because the denominator \( \ln x \) approaches zero as \( x \) approaches 1 from the right. Analyze this improper behavior at \( x = 1 \).

Step 2: Set Up the Limit to Handle the Impropriety

Since \( \ln x \to 0^+ \) as \( x \to 1^+ \), rewrite the integral using a limit to manage this impropriety:\[ \int_{1}^{2} \frac{d x}{x \ln x} = \lim_{a \to 1^+} \int_{a}^{2} \frac{d x}{x \ln x}. \]

Step 3: Use Substitution to Simplify the Integral

Use the substitution \( u = \ln x \), which implies \( du = \frac{1}{x} dx \). The limits of integration change from \( x=a \) to \( u=\ln a \) and from \( x=2 \) to \( u=\ln 2 \). Substituting, we have:\[ \lim_{a \to 1^+} \int_{\ln a}^{\ln 2} \frac{1}{u} \, du. \]

Step 4: Solve the Simplified Integral

The antiderivative of \( \frac{1}{u} \) is \( \ln |u| \). Evaluate the definite integral:\[ \lim_{a \to 1^+} [\ln |u|]_{\ln a}^{\ln 2} = \lim_{a \to 1^+} (\ln (\ln 2) - \ln (\ln a)). \]

Step 5: Evaluate the Limit

As \( a \to 1^+ \), \( \ln a \to 0^+ \), which makes \( \ln(\ln a) \to -\infty \). Thus, the limit becomes:\[ \ln (\ln 2) - (-\infty) = +\infty. \]

Step 6: Conclusion on the Convergence

Since the evaluated limit results in \(+\infty\), the integral \( \int_{1}^{2} \frac{d x}{x \ln x} \) diverges.

Key Concepts

Convergence

In calculus, when we talk about the convergence of an integral, we are interested in whether the integral has a finite value or not. For an integral to converge, the overall area under the curve must approach a finite value as the limit of integration reaches its specified boundary.

• In improper integrals, where the integrand might become undefined at some points, convergence becomes a critical concept.
• If the integral converges, it means the limit exists and yields a real number.
• Conversely, if the area under the curve stretches to infinity, the integral is said to diverge.

Evaluating convergence is essential because it tells us about the behavior of the function over the specified interval.

Substitution Method

The substitution method is a powerful technique used to simplify and solve integrals. This method often makes complex integrals more approachable by transforming them into a form that is easier to evaluate.

• By substituting a part of the integrand with a single variable, substitution can change the problem into a simpler standard form.
• In our exercise, we used the substitution \( u = \ln x \), which lets us convert \( \int \frac{dx}{x \ln x} \) into \( \int \frac{1}{u} du \).
• This transformation simplified the integral by removing the dependency on \( x \), allowing us to work with a more familiar antiderivative.

Mastering substitution can significantly ease solving both definite and indefinite integrals.

Limit Evaluation

Evaluating limits is a fundamental tool in the study of improper integrals. To handle points where the function is undefined or infinite, limits allow us to approach these problematic points cautiously.

• This method involves redefining the integral in terms of a limit approaching the troublesome boundary.
• For the integral in the exercise, the limit \( \lim_{a \to 1^+} \int_{a}^{2} \frac{dx}{x \ln x} \) was set to handle the point \( x=1 \) where \( \ln x = 0 \).
• By evaluating this limit, we determine whether the behavior around the point leads us to a finite result or not.

Limit evaluation helps bridge gaps where direct integration isn’t possible due to the function's behavior.

Integral Diverges

When an integral diverges, it indicates that the area under the curve is infinite and not bounded. This means that the integral does not converge to a finite number.

• A divergent integral tells us that the function grows without bound over the interval, or around a particular point in the interval.
• In our example, as we evaluated \( \lim_{a \to 1^+} (\ln(\ln 2) - \ln(\ln a)) \), the result was \(+\infty\).
• This outcome confirms that the integral \( \int_{1}^{2} \frac{d x}{x \ln x} \) stretches towards infinity, diverging as a result.

Recognizing divergence is crucial, as it indicates significant behavior in the function that may be problematic or unsolvable in the usual sense.