To evaluate the integral:

$\int (1 + 3x^3) e^{x^3} \, dx$

Step-by-Step Solution:

1. Recognize a substitution pattern:
   The derivative of $x^3$ is $3x^2$, and the integrand contains terms that suggest substitution with $u = x^3$.

2. Substitute: Let: $u = x^3 \implies du = 3x^2 dx$ This simplifies $dx$ as: $dx = \frac{du}{3x^2}$

3. Rewrite the integral:
   Substitute $u$ and adjust the terms: $\int (1 + 3x^3) e^{x^3} dx = \int \left(1 + 3u\right) e^u \frac{du}{3x^2}$ However, $x^2$ remains in the denominator. From $u = x^3$, we know $x^2 = u^{2/3}$. Substituting this: $dx = \frac{du}{3u^{2/3}}$ So the integral becomes: $\int (1 + 3u)e^u \frac{1}{3u^{2/3}} du = \frac{1}{3} \int \frac{(1 + 3u)e^u}{u^{2/3}} du$

4. Simplify further and evaluate:
   The resulting integral is a combination of exponential and fractional powers, which may need a series expansion for further resolution or numeric methods if closed forms aren't possible. Let me know if you'd like a deeper step into series or numeric computation!

Would you like me to refine this for approximation, or proceed with series expansion?

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Follow-Up Questions:

1. Do you want to explore methods for solving integrals with fractional exponents?
2. Should I demonstrate a numeric approximation for this integral?
3. Would you like to explore how substitution simplifies complex exponential integrals?
4. Are you curious about series expansion for functions with $e^u$?
5. Should we generalize this problem for powers other than $x^3$?

Tip:

For integrals involving products of polynomials and exponentials, substitution is often the first step, but not always sufficient for a closed-form solution.