1. **Problem:** Express the integral $$\int_0^1 (2x - 1)^3 \, dx$$ in terms of the variable $$u = 2x - 1$$ without evaluating it. 2. **Formula and substitution rule:** When substituting $$u = g(x)$$, we use $$du = g'(x) dx$$. The limits of integration change accordingly: if $$x = a$$, then $$u = g(a)$$; if $$x = b$$, then $$u = g(b)$$. 3. **Apply substitution:** Given $$u = 2x - 1$$, then $$\frac{du}{dx} = 2$$ or $$dx = \frac{du}{2}$$. 4. **Change limits:** When $$x = 0$$, $$u = 2(0) - 1 = -1$$. When $$x = 1$$, $$u = 2(1) - 1 = 1$$. 5. **Rewrite integral:** $$\int_0^1 (2x - 1)^3 \, dx = \int_{u=-1}^{u=1} u^3 \cdot \frac{du}{2} = \frac{1}{2} \int_{-1}^1 u^3 \, du$$ **Final answer:** $$\frac{1}{2} \int_{-1}^1 u^3 \, du$$