1. **Problem:** Evaluate the integral $$\int (2x - 3) \sqrt[4]{x - 2} \, dx$$ using substitution. 2. **Step 1: Identify substitution.** Let $$u = x - 2$$ so that $$du = dx$$ and $$x = u + 2$$. 3. **Rewrite the integral in terms of $$u$$:** $$\int (2(u + 2) - 3) u^{1/4} \, du = \int (2u + 4 - 3) u^{1/4} \, du = \int (2u + 1) u^{1/4} \, du$$ 4. **Distribute $$u^{1/4}$$:** $$\int (2u \cdot u^{1/4} + 1 \cdot u^{1/4}) \, du = \int (2u^{5/4} + u^{1/4}) \, du$$ 5. **Integrate term-by-term:** $$\int 2u^{5/4} \, du + \int u^{1/4} \, du = 2 \int u^{5/4} \, du + \int u^{1/4} \, du$$ 6. **Use power rule for integration:** $$\int u^n \, du = \frac{u^{n+1}}{n+1} + C$$ 7. **Calculate each integral:** $$2 \cdot \frac{u^{5/4 + 1}}{5/4 + 1} + \frac{u^{1/4 + 1}}{1/4 + 1} + C = 2 \cdot \frac{u^{9/4}}{9/4} + \frac{u^{5/4}}{5/4} + C$$ 8. **Simplify fractions:** $$2 \cdot \frac{4}{9} u^{9/4} + \frac{4}{5} u^{5/4} + C = \frac{8}{9} u^{9/4} + \frac{4}{5} u^{5/4} + C$$ 9. **Substitute back $$u = x - 2$$:** $$\boxed{\frac{8}{9} (x - 2)^{9/4} + \frac{4}{5} (x - 2)^{5/4} + C}$$ This is the evaluated integral.