Let's solve **Question (1.1)** step-by-step: --- ## (1.1) \(\int \frac{(\sqrt{x} - 4)^2}{\sqrt[3]{x}} \, dx\) ### **Step 1: Expand the numerator** \[ (\sqrt{x} - 4)^2 = (\sqrt{x})^2 - 2 \cdot 4 \cdot \sqrt{x} + 16 = x - 8\sqrt{x} + 16 \] So, \[ \int \frac{x - 8\sqrt{x} + 16}{\sqrt[3]{x}} \, dx \] --- ### **Step 2: Write each term separately** Recall: \(\sqrt{x} = x^{1/2}\) and \(\sqrt[3]{x} = x^{1/3}\): \[ = \int \frac{x}{x^{1/3}} \, dx - 8 \int \frac{x^{1/2}}{x^{1/3}} \, dx + 16 \int \frac{1}{x^{1/3}} \, dx \] --- ### **Step 3: Simplify each term** - \(\frac{x}{x^{1/3}} = x^{1 - 1/3} = x^{2/3}\) - \(\frac{x^{1/2}}{x^{1/3}} = x^{1/2 - 1/3} = x^{(3-2)/6} = x^{1/6}\) - \(\frac{1}{x^{1/3}} = x^{-1/3}\) So the integral becomes: \[ \int x^{2/3} dx - 8 \int x^{1/6} dx + 16 \int x^{-1/3} dx \] --- ### **Step 4: Integrate each term** Recall: \[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \] - \(\int x^{2/3} dx = \frac{x^{5/3}}{5/3} = \frac{3}{5} x^{5/3}\) - \(\int x^{1/6} dx = \frac{x^{7/6}}{7/6} = \frac{6}{7} x^{7/6}\) - \(\int x^{-1/3} dx = \frac{x^{2/3}}{2/3} = \frac{3}{2} x^{2/3}\) --- ### **Step 5: Substitute back** \[ = \frac{3}{5}x^{5/3} - 8 \cdot \frac{6}{7} x^{7/6} + 16 \cdot \frac{3}{2} x^{2/3} + C \] \[ = \frac{3}{5}x^{5/3} - \frac{48}{7} x^{7/6} + 24 x^{2/3} + C \] --- ## **Final Answer:** \[ \boxed{ \int \frac{(\sqrt{x} - 4)^2}{\sqrt[3]{x}} dx = \frac{3}{5}x^{5/3} - \frac{48}{7}x^{7/6} + 24x^{2/3} + C } \] --- Would you like the next integral solved as well?