To find the integral of the expression $-x^2 + 1$, we need to integrate each term separately.

Integral:

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

We can split the integral:

$\int -x^2 \, dx + \int 1 \, dx$

Now, integrate each term:

1. For $\int -x^2 \, dx$, the power rule gives:

$\int -x^2 \, dx = -\frac{x^3}{3}$

2. For $\int 1 \, dx$, the integral of a constant is:

$\int 1 \, dx = x$

So, putting them together:

$\int (-x^2 + 1) \, dx = -\frac{x^3}{3} + x + C$

where $C$ is the constant of integration.

Final Answer:

$-\frac{x^3}{3} + x + C$

Would you like to dive deeper into any of the steps or concepts? Here are some related questions you might find interesting:

1. What is the power rule for integration?
2. How do we handle constants when integrating?
3. Can you explain the concept of an indefinite integral in more detail?
4. How do definite integrals differ from indefinite integrals?
5. What is the role of the constant of integration in indefinite integrals?

Tip: When integrating polynomials, always add 1 to the exponent and divide by the new exponent, unless the exponent is $-1$ (in which case it's a logarithm).