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.

Learn how to integrate x times (1 plus x squared) with detailed step-by-step solutions. This problem involves integrating polynomial functions and requires understanding the rules of integration, suitable for advanced high school students.

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