The image asks to evaluate the integral

$\int_1^\infty e^{-x^2} \, dx$

and determine whether it is divergent or convergent.

Evaluation:

The integral of the Gaussian function $e^{-x^2}$ over a finite interval is not solvable in terms of elementary functions. However, for limits that extend to infinity, it’s useful to understand the behavior of the function.

Convergence:

The integrand $e^{-x^2}$ decays very rapidly as $x \to \infty$, because as $x$ increases, $e^{-x^2}$ approaches 0 extremely quickly. To determine whether the integral converges, we can analyze the improper integral:

$\int_1^\infty e^{-x^2} \, dx$

Since $e^{-x^2}$ decays faster than any polynomial as $x \to \infty$, this integral converges. A similar integral over $(-\infty, \infty)$, known as the Gaussian integral, is known to converge, and thus, this integral also converges on $[1, \infty)$.

Conclusion:

The integral $\int_1^\infty e^{-x^2} \, dx$ converges.

Would you like me to go into more details or have any questions?

Here are 5 related questions to expand your knowledge:

1. How would you evaluate the Gaussian integral over the entire real line?
2. Can you express $\int_0^\infty e^{-x^2} \, dx$ using known special functions?
3. How do you determine the convergence of an improper integral?
4. What techniques can be used to evaluate integrals that involve exponential decay?
5. What role does the error function $\text{erf}(x)$ play in integrals involving $e^{-x^2}$?

Tip: When dealing with improper integrals, it's essential to assess the behavior of the function at the limits of integration (e.g., as $x \to \infty$ or at singular points).