Evaluate the definite integral: ∫₀¹ x(x² - 2)¹ dx

The problem asks to evaluate the definite integral:

$$\int_{0}^{1} x(x^2 - 2)^1 dx$$

The exponent on $(x^2 - 2)$ is interpreted as 1 due to the image clarity.

Step 1: Simplify the integrand.

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

Step 2: Find the antiderivative of the simplified integrand.

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