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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.

Consider the function f(x) defined by f(x) = ∫[0, x] (e^(t^2)) / (1 + t^2) dt.

This problem involves the integral function f(x) = ∫[0, x] (e^(t^2)) / (1 + t^2) dt, exploring its derivative using the Fundamental Theorem of Calculus. A step-by-step explanation for finding f'(x) and analyzing its behavior is provided. Suitable for advanced calculus students.

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