Question

Evaluate each improper integral or show that it diverges. \(\int_{e}^{\infty} \frac{1}{x \ln x} d x\)

Short answer

The integral diverges.

Step-by-step solution

Write the Improper Integral in Limit Form

The improper integral from \( e \) to \( \infty \) for the function \( \frac{1}{x \ln x} \) is expressed using limits: \[ \int_{e}^{\infty} \frac{1}{x \ln x} \, dx = \lim_{b \to \infty} \int_{e}^{b} \frac{1}{x \ln x} \, dx. \]

Use Substitution Method

Let \( u = \ln x \). Then, \( \frac{du}{dx} = \frac{1}{x} \), or \( du = \frac{1}{x} dx \). Rewriting the integral in terms of \( u \), we have: \[ \int \frac{1}{x \ln x} \cdot x \, du = \int \frac{1}{u} \, du. \]

Evaluate the Integral

The integral of \( \frac{1}{u} \, du \) is \( \ln |u| + C \). So, the integral becomes \( \ln |\ln x| + C \). Apply the limits: \[ \lim_{b \to \infty} \int_{e}^{b} \frac{1}{x \ln x} \, dx = \lim_{b \to \infty} [ \ln |\ln x| ]_{x=e}^{x=b}. \]

Evaluate the Limit

Substitute the limits into the evaluated integral: \[ \lim_{b \to \infty} ( \ln |\ln b| - \ln |\ln e| ). \] Since \( \ln e = 1 \), this simplifies to \[ \lim_{b \to \infty} ( \ln |\ln b| - \ln(1) ) = \lim_{b \to \infty} \ln |\ln b|. \] As \( b \to \infty \), \( \ln b \) also approaches infinity, and so does \( \ln |\ln b| \).

Conclusion

Since \( \ln |\ln b| \to \infty \) as \( b \to \infty \), the integral \( \int_{e}^{\infty} \frac{1}{x \ln x} \, dx \) diverges.

Key concepts

Divergence of Integrals

In the realm of calculus, improper integrals often challenge us with boundaries that stretch to infinity. A key aspect of understanding these integrals is determining whether they converge to a finite value or diverge, spiraling off into infinity.
For the integral \[ \int_{e}^{\infty} \frac{1}{x \ln x} \, dx, \] we need to assess the behavior as the upper limit tends towards infinity. As we identify the function within the integral, it's evident that as \( x \) increases, \( \ln x \) also increases, making \( \frac{1}{x \ln x} \) tend towards zero.
However, despite this diminishing value, if the rate of decay is not fast enough, the area under the curve could still be infinite, indicating divergence.
The conclusion is that the improper integral diverges as \( \ln |\ln b| \) becomes infinitely large.

Integration Techniques

Handling improper integrals requires employing specific integration techniques that allow us to simplify the problem. These techniques can transform a seemingly complex integral into a more manageable form.
A crucial skill is recognizing when to apply substitution, a technique that often simplifies integrals by changing variables. This integral, \[ \int \frac{1}{x \ln x} \, dx, \]is simplified through substitution. By selecting \( u = \ln x \), we can reduce the integral's complexity. This technique casts the integral into a different light, making it easier to work with by integrating \( \frac{1}{u} \) instead of the original function.
Such transformations illustrate the power of substitution in unlocking the solution to more complex integrals.

Limit Evaluation

Improper integrals often involve evaluating limits to assess convergence or divergence. For the given integral, the limit plays a crucial role in determining the integral's behavior as the upper boundary approaches infinity.
After rewriting the integral using a limit,
\[ \lim_{b \to \infty} \int_{e}^{b} \frac{1}{x \ln x} \, dx, \]the problem simplifies to examining the expression
\[ \lim_{b \to \infty} ( \ln |\ln b| - \ln |\ln e| ).\]
As \( b \to \infty \), \( \ln b \) also increases without bounds, revealing the divergence of the integral.
This approach captures the essence of limit evaluation as not only a process of substitution but also as an overarching check for convergence.

Substitution Method

The substitution method is an essential technique in calculus used to simplify the integration process. This involves replacing a complex part of the integral with a single variable, transforming it into an easier form to solve.
In the example, the substitution \( u = \ln x \) offers a pathway to simplify \[ \int \frac{1}{x \ln x} \, dx \] to \[ \int \frac{1}{u} \, du. \] This substitution is possible because the derivative of \( \ln x \) corresponds to \( \frac{1}{x} \). Consequently, the dx part of the integral is expressed as \( du \).
This conversion allows for straightforward integration, yielding \( \ln |u| \) plus a constant. In original variables, this converts back to \( \ln |\ln x| \).
Through substitution, we not only make computation easier but also draw a clearer path to evaluating or recognizing divergence in an improper integral.