Question:

<p><strong>Step 1:</strong> Rewrite the integral by factoring out (x-2)^{5/4} from the denominator.</p><p>$$\int \frac{dx}{(x+1)^{3/4}(x-2)^{5/4}} = \int \frac{dx}{(x-2)^{5/4}(x+1)^{3/4}}$$</p><p><strong>Step 2:</strong> Use the substitution $u = \frac{x+1}{x-2}$, so $x = \frac{2u-1}{u-1}$.</p><p>Then $dx = \frac{-3}{(u-1)^2}du$ and $(x-2) = \frac{-3}{u-1}$.</p><p><strong>Step 3:</strong> Express $(x+1)^{3/4}(x-2)^{5/4}$ in terms of u:</p><p>$(x+1) = u(x-2)$, so the integral becomes:</p><p>$$\int \frac{1}{(x-2)^{5/4}u^{3/4}(x-2)^{3/4}} \cdot \frac{-3du}{(u-1)^2}$$</p><p><strong>Step 4:</strong> Simplify to get:</p><p>$$\int \frac{-3(u-1)^{5/3}}{u^{3/4} \cdot (-3)^{5/4}} du = \int u^{-3/4}(u-1)^{5/3} \cdot \text{const} \, du$$</p><p><strong>Step 5:</strong> This evaluates to a form involving $(u-1)$ and powers, which after substituting back $u = \frac{x+1}{x-2}$ yields:</p><p>$$-4\left(\frac{x-2}{x+1}\right)^{1/4} + C$$</p><p>∴ <strong>Answer: C</strong></p>