The given problem defines the function $f(x)$ as follows:

$f(x) = \int_0^x \frac{e^{t^2}}{1 + t^2} \, dt.$

To analyze this function, we can take various steps, such as:

1. Finding $f'(x)$ using the Fundamental Theorem of Calculus: The derivative of $f(x)$ can be found directly from the Leibniz rule, which states:

   $f'(x) = \frac{d}{dx} \left( \int_0^x g(t) \, dt \right) = g(x),$

   where $g(t)$ is the integrand. In this case:

   $f'(x) = \frac{e^{x^2}}{1 + x^2}.$

2. Analyzing the behavior of $f(x)$:

   • Since the integrand $\frac{e^{t^2}}{1 + t^2}$ is always positive, the function $f(x)$ is increasing.
   • As $x \to \infty$, the exponential term $e^{x^2}$ grows very rapidly, suggesting that $f(x) \to \infty$ as $x \to \infty$.
   • As $x \to 0$, the integral vanishes, so $f(0) = 0$.

Would you like me to go into further detail on any specific part of the function or explore other properties of $f(x)$?

Here are five related questions:

1. How can we determine whether $f(x)$ is concave or convex?
2. What is the behavior of $f(x)$ as $x \to \infty$?
3. Can we approximate $f(x)$ using numerical methods?
4. How can we evaluate the definite integral of this function at specific points?
5. How does the integrand $\frac{e^{t^2}}{1 + t^2}$ affect the overall growth of $f(x)$?

Tip: The Fundamental Theorem of Calculus is a powerful tool for calculating derivatives of integrals with variable limits.