To solve the integral 

\[
I = \int \frac{x^3 e^{x^2}}{(x^2 + 1)^2} \, dx,
\]

let's try substitution to simplify the expression.

### Step 1: Substitution
Let \( u = x^2 + 1 \), which implies that:

\[
du = 2x \, dx \quad \text{or} \quad \frac{du}{2} = x \, dx.
\]

Now, observe that \( x^2 = u - 1 \), so:

\[
x^3 = x \cdot x^2 = x(u - 1).
\]

Thus, the integral becomes:

\[
I = \int \frac{x(u - 1) e^{u - 1}}{u^2} \cdot \frac{du}{2}.
\]

### Step 2: Simplifying the Integral
We can now factor constants and simplify:

\[
I = \frac{1}{2} \int \frac{x(u - 1) e^{u - 1}}{u^2} \, du.
\]

Let’s now evaluate or proceed with any further simplification.

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### Here are 5 follow-up questions for you:
1. Do you want to continue with a deeper look into integration techniques for this?
2. Would you like to see an alternative method to solve this integral?
3. Should we explore the limits of this integral for specific values of \(x\)?
4. Are you interested in understanding the behavior of the function as \( x \to \infty \)?
5. Would you like help in computing this integral numerically?

**Tip:** Substitutions are often helpful in integrals when the inner function in a composite function can simplify both the expression and its derivative.

\int \frac{x^3 e^{x^2}}{(x^2 + 1)^2} \, dx

This page walks through solving the integral \int \frac{x^3 e^{x^2}}{(x^2 + 1)^2} using substitution and exponential function techniques. Key concepts include the substitution method, exponential functions, and rational expressions, suitable for college-level calculus students.

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