1. **Problem statement:** Calculate the definite integral $\int_{-1}^1 (x^3 - 2x + 1) \, dx$. 2. **Formula and rules:** The definite integral of a function $f(x)$ from $a$ to $b$ is given by $$\int_a^b f(x) \, dx = F(b) - F(a)$$ where $F(x)$ is any antiderivative (stammfunktion) of $f(x)$. 3. **Find the antiderivative:** Given $f(x) = x^3 - 2x + 1$, integrate term-by-term: $$F(x) = \frac{x^4}{4} - x^2 + x + C$$ 4. **Evaluate the definite integral:** $$\int_{-1}^1 (x^3 - 2x + 1) \, dx = F(1) - F(-1)$$ Calculate $F(1)$: $$F(1) = \frac{1^4}{4} - 1^2 + 1 = \frac{1}{4} - 1 + 1 = \frac{1}{4}$$ Calculate $F(-1)$: $$F(-1) = \frac{(-1)^4}{4} - (-1)^2 + (-1) = \frac{1}{4} - 1 - 1 = \frac{1}{4} - 2 = -\frac{7}{4}$$ 5. **Subtract to find the integral value:** $$F(1) - F(-1) = \frac{1}{4} - \left(-\frac{7}{4}\right) = \frac{1}{4} + \frac{7}{4} = 2$$ **Final answer:** $$\int_{-1}^1 (x^3 - 2x + 1) \, dx = 2$$