\( \left. \begin{array} { l } { \frac { 4 } { 5 } \cdot \sqrt { x } \cdot x + \int 32 x = \frac { 4 \sqrt { 2 } } { 5 } + 3 | 3 } \\ { \int _ { 3 } ^ { 5 } \frac { 4 \sqrt { 5 } \cdot | 3 | } { 5 } + 3 \cdot 3 - ( \frac { \sqrt[ 4 ] { 3 } \cdot | 3 | } { 5 } + 3 \cdot 3 } \end{array} \right. \)

To solve the expression you've provided, we need to break it down into manageable parts. Let's analyze each component step by step. 1. **First Expression**: \[ \frac{4}{5} \cdot \sqrt{x} \cdot x + \int 32x \, dx = \frac{4\sqrt{2}}{5} + 3|3| \] - The integral \(\int 32x \, dx\) can be computed as follows: \[ \int 32x \, dx = 16x^2 + C \] - Therefore, the left side becomes: \[ \frac{4}{5} \cdot \sqrt{x} \cdot x + 16x^2 + C \] - The right side simplifies to: \[ \frac{4\sqrt{2}}{5} + 3 \cdot 3 = \frac{4\sqrt{2}}{5} + 9 \] - So we have: \[ \frac{4}{5} \cdot \sqrt{x} \cdot x + 16x^2 + C = \frac{4\sqrt{2}}{5} + 9 \] 2. **Second Expression**: \[ \int_{3}^{5} \frac{4\sqrt{5} \cdot |3|}{5} + 3 \cdot 3 - \left( \frac{\sqrt[4]{3} \cdot |3|}{5} + 3 \cdot 3 \right) \] - First, calculate \(|3|\): \[ |3| = 3 \] - Substitute \(|3|\) into the expression: \[ \int_{3}^{5} \frac{4\sqrt{5} \cdot 3}{5} + 9 - \left( \frac{\sqrt[4]{3} \cdot 3}{5} + 9 \right) \] - The integral simplifies to: \[ \int_{3}^{5} \frac{12\sqrt{5}}{5} \, dx = \frac{12\sqrt{5}}{5} \cdot (5 - 3) = \frac{12\sqrt{5}}{5} \cdot 2 = \frac{24\sqrt{5}}{5} \] - Now, the entire expression becomes: \[ \frac{24\sqrt{5}}{5} + 9 - \left( \frac{3\sqrt[4]{3}}{5} + 9 \right) \] - The \(9\) terms cancel out: \[ \frac{24\sqrt{5}}{5} - \frac{3\sqrt[4]{3}}{5} \] - Factor out \(\frac{1}{5}\): \[ \frac{1}{5} \left( 24\sqrt{5} - 3\sqrt[4]{3} \right) \] 3. **Final Result**: The final result of the entire expression is: \[ \frac{1}{5} \left( 24\sqrt{5} - 3\sqrt[4]{3} \right) \] This is the simplified form of the expression you provided.