`int_0^(64)[{x^(1//3)}-{sqrtx}]dx = .....`

Answer

To solve the integral \( \int_0^{64} \left( x^{\frac{1}{3}} - \sqrt{x} \right) dx \), we will follow these steps:

### Step 1: Set up the integral
We need to evaluate the integral from 0 to 64 of the function \( x^{\frac{1}{3}} - \sqrt{x} \).

\[ \int_0^{64} \left( x^{\frac{1}{3}} - x^{\frac{1}{2}} \right) dx \]

### Step 2: Integrate each term separately
We will integrate \( x^{\frac{1}{3}} \) and \( -\sqrt{x} \) separately.

1. **Integrate \( x^{\frac{1}{3}} \)**:
\[ \int x^{\frac{1}{3}} dx = \frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4} x^{\frac{4}{3}} \]

2. **Integrate \( -\sqrt{x} \)**:
\[ \int -\sqrt{x} dx = -\int x^{\frac{1}{2}} dx = -\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} = -\frac{x^{\frac{3}{2}}}{\frac{3}{2}} = -\frac{2}{3} x^{\frac{3}{2}} \]

### Step 3: Combine the results
Now we combine the results of the integrals:
\[ \int \left( x^{\frac{1}{3}} - \sqrt{x} \right) dx = \frac{3}{4} x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} \]

### Step 4: Evaluate the definite integral from 0 to 64
Now we will evaluate the combined integral from 0 to 64:
\[ \left[ \frac{3}{4} x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} \right]_0^{64} \]

1. **Evaluate at the upper limit (64)**:
\[ \frac{3}{4} (64)^{\frac{4}{3}} - \frac{2}{3} (64)^{\frac{3}{2}} \]

- Calculate \( (64)^{\frac{4}{3}} \):
\[ (64)^{\frac{4}{3}} = (4^3)^{\frac{4}{3}} = 4^4 = 256 \]
- Calculate \( (64)^{\frac{3}{2}} \):
\[ (64)^{\frac{3}{2}} = (8^2)^{\frac{3}{2}} = 8^3 = 512 \]

Now substituting these values:
\[ \frac{3}{4} \cdot 256 - \frac{2}{3} \cdot 512 = 192 - \frac{1024}{3} \]

2. **Evaluate at the lower limit (0)**:
\[ \frac{3}{4} (0)^{\frac{4}{3}} - \frac{2}{3} (0)^{\frac{3}{2}} = 0 \]

### Step 5: Final calculation
Now we combine the results:
\[ 192 - \frac{1024}{3} = \frac{576}{3} - \frac{1024}{3} = \frac{576 - 1024}{3} = \frac{-448}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_0^{64} \left( x^{\frac{1}{3}} - \sqrt{x} \right) dx = -\frac{448}{3} \]

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