$$ \int_0^{\infty}\!\frac{1}{x^2}\,\mathrm{d}x $$

The integral you're asking about is:

\[ \int_0^{\infty} \frac{1}{x^2} \, \mathrm{d}x \]

Let's analyze this integral. 

First, we rewrite the integrand:

\[ \frac{1}{x^2} = x^{-2} \]

Thus, the integral becomes:

\[ \int_0^{\infty} x^{-2} \, \mathrm{d}x \]

To evaluate this improper integral, we need to check its behavior at the boundaries:

1. Near $ x = 0 $
2. As $ x \to \infty $

### Behavior Near $ x = 0 $

\[ \int_0^{a} x^{-2} \, \mathrm{d}x \]

\[ = \left[ \frac{x^{-1}}{-1} \right]_0^a \]

\[ = -\left[ \frac{1}{x} \right]_0^a \]

\[ = -\left( \frac{1}{a} - \lim_{x \to 0} \frac{1}{x} \right) \]

The limit as $ x \to 0 $ is undefined (it goes to $\infty$), so the integral diverges at the lower bound.

### Behavior As $ x \to \infty $

\[ \int_a^{\infty} x^{-2} \, \mathrm{d}x \]

\[ = \left[ \frac{x^{-1}}{-1} \right]_a^\infty \]

\[ = -\left[ \frac{1}{x} \right]_a^\infty \]

\[ = -\left( \lim_{x \to \infty} \frac{1}{x} - \frac{1}{a} \right) \]

\[ = -\left( 0 - \frac{1}{a} \right) \]

\[ = \frac{1}{a} \]

Thus, the integral converges as $ x \to \infty $.

Since the integral diverges at the lower bound ($ x = 0 $), the overall integral:

\[ \int_0^{\infty} \frac{1}{x^2} \, \mathrm{d}x \]

is divergent.

Would you like more details or have any questions about this?

Here are five related questions you might consider:

1. How do you evaluate improper integrals with infinite limits?
2. What are the conditions for an integral to converge?
3. Can you provide examples of other divergent integrals?
4. How does one deal with integrals that diverge at both bounds?
5. What are some techniques to determine the convergence of an integral?

**Tip:** When dealing with improper integrals, always check the behavior of the integrand at the boundaries of integration to determine if the integral converges or diverges.

Explore the improper integral ∫(1/x^2)dx from 0 to ∞. Understand why this integral diverges at the lower bound and converges at the upper bound through detailed analysis. Suitable for advanced college-level calculus students.

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