**2. Determine the convergence of the improper integral**\[\int_{0}^{\infty} e^{x} \ln x \, dx.\]In this problem, we are tasked with determining whether the given improper integral converges or diverges. The integral is defined over the interval from 0 to infinity, and the function to be integrated is $ e^{x} \ln x $. To evaluate convergence, one typically considers the behavior of the function as $ x $ approaches both the lower and upper bounds of the integration interval. Specifically, we need to examine the behavior near $ x = 0 $ and as $ x \to \infty $ to determine if the area under the curve is finite or infinite.### Steps to Analyze1. **Behavior as $ x \to 0^+ $:** - Consider if $ \ln x $ poses any issues in terms of singularity and if the resulting function $ e^x \ln x $ is manageable around this limit.2. **Behavior as $ x \to \infty $:** - Assess the dominance of the exponential growth due to $ e^x $ over the function and whether it results in divergence at the upper limit of integration.### Explanation and Analysis:- **Graphically**, one might imagine $ e^x \ln x $ starting from zero and potentially rising rapidly due to the exponential component, depending on dominance and interaction with $ \ln x $.- A more advanced mathematical approach may involve using integral convergence tests or specific techniques for handling improper integrals like comparison tests or transformations.The conclusion involves stating whether the integral converges or diverges based on a thorough theoretical or calculated assessment.

Take the integral, ∫0∞xlnx=lnxx22-∫0∞1x.x22dx=x2lnx2-x240∞=∞ As after taking x→∞ the limit of the…

Answered: 2. Determine the convergence of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**2. Determine the convergence of the improper integral**

\[
\int_{0}^{\infty} e^{x} \ln x \, dx.
\]

In this problem, we are tasked with determining whether the given improper integral converges or diverges. The integral is defined over the interval from 0 to infinity, and the function to be integrated is $ e^{x} \ln x $.

To evaluate convergence, one typically considers the behavior of the function as $ x $ approaches both the lower and upper bounds of the integration interval. Specifically, we need to examine the behavior near $ x = 0 $ and as $ x \to \infty $ to determine if the area under the curve is finite or infinite.

### Steps to Analyze

1. **Behavior as $ x \to 0^+ $:**
- Consider if $ \ln x $ poses any issues in terms of singularity and if the resulting function $ e^x \ln x $ is manageable around this limit.

2. **Behavior as $ x \to \infty $:**
- Assess the dominance of the exponential growth due to $ e^x $ over the function and whether it results in divergence at the upper limit of integration.

### Explanation and Analysis:

- **Graphically**, one might imagine $ e^x \ln x $ starting from zero and potentially rising rapidly due to the exponential component, depending on dominance and interaction with $ \ln x $.
- A more advanced mathematical approach may involve using integral convergence tests or specific techniques for handling improper integrals like comparison tests or transformations.

The conclusion involves stating whether the integral converges or diverges based on a thorough theoretical or calculated assessment.

Expert Solution

Step 1

Take the integral,

$\begin{array}{rcl} \int_{0}^{\infty} x \ln x & = & \ln x (\frac{x^{2}}{2}) - \int_{0}^{\infty} \frac{1}{x} . \frac{x^{2}}{2} d x \\ = & {[\frac{x^{2} \ln x}{2} - \frac{x^{2}}{4}]}_{0}^{\infty} \\ = & \infty \end{array}$

As after taking $x \to \infty$ the limit of the integral will be:

$\lim_{x \to \infty} \int_{0}^{\infty} x \ln x d x = \infty$

The integral will diverge.

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