The integral you provided looks like:

$\int -2x(1 - x^2) \, dx$

Let's solve it step by step:

Step 1: Expand the expression inside the integral.

We can distribute $-2x$ across $(1 - x^2)$:

$-2x(1 - x^2) = -2x + 2x^3$

So the integral becomes:

$\int (-2x + 2x^3) \, dx$

Step 2: Integrate each term.

• The integral of $-2x$ is $-x^2$.
• The integral of $2x^3$ is $\frac{1}{2} \cdot 2x^4 = \frac{1}{2}x^4$.

Thus, the integral becomes:

$-x^2 + \frac{1}{2}x^4 + C$

where $C$ is the constant of integration.

Final Answer:

$-x^2 + \frac{1}{2}x^4 + C$

Would you like more details or have any questions? Here are some related questions you can explore:

1. How do you integrate polynomials in general?
2. What is the power rule of integration, and why does it work?
3. How do you handle negative exponents in integrals?
4. How does integration relate to the area under a curve?
5. How would this problem change if we had definite limits of integration?

Tip: Always double-check your distribution when expanding expressions to avoid calculation errors!