To solve the integral

$\int x \cdot (1 + x^2) \, dx,$

we can simplify the expression first and then integrate term by term. Let's rewrite the integrand:

$x \cdot (1 + x^2) = x + x^3.$

Thus, the integral becomes:

$\int (x + x^3) \, dx.$

Now, we integrate each term separately:

$\int x \, dx + \int x^3 \, dx.$

The integral of $x$ with respect to $x$ is:

$\int x \, dx = \frac{x^2}{2}.$

The integral of $x^3$ with respect to $x$ is:

$\int x^3 \, dx = \frac{x^4}{4}.$

Putting these together, we get:

$\int (x + x^3) \, dx = \frac{x^2}{2} + \frac{x^4}{4} + C,$

where $C$ is the constant of integration.

Therefore, the solution to the integral is:

$\boxed{\frac{x^2}{2} + \frac{x^4}{4} + C}.$

Would you like any details or have any questions?

Here are 5 numbered questions you can ask:

1. How do you integrate $x \cdot e^x$?
2. Can you explain the method of integration by parts?
3. How do you solve a definite integral from 0 to 1 of $x^2$?
4. What is the integral of $\sin(x) \cdot \cos(x)$?
5. How do you find the area under a curve using integration?

Tip: When integrating polynomial functions, break them down into simpler terms that can be integrated individually.