To evaluate the definite integral:

$\int_{2}^{8} \frac{6x^2 + 8}{\sqrt{x}} \, dx$

Step 1: Simplify the integrand

Split the numerator as:

$\frac{6x^2 + 8}{\sqrt{x}} = \frac{6x^2}{\sqrt{x}} + \frac{8}{\sqrt{x}} = 6x^{3/2} + 8x^{-1/2}$

Step 2: Write the integral in terms of these simplified expressions

The integral becomes:

$\int_{2}^{8} \left( 6x^{3/2} + 8x^{-1/2} \right) \, dx$

Step 3: Integrate term by term

Use the power rule for integration, which states:

$\int x^n \, dx = \frac{x^{n+1}}{n+1}, \quad n \neq -1$

First term:

$\int 6x^{3/2} \, dx = 6 \cdot \frac{x^{5/2}}{5/2} = \frac{12}{5} x^{5/2}$

Second term:

$\int 8x^{-1/2} \, dx = 8 \cdot \frac{x^{1/2}}{1/2} = 16x^{1/2}$

Thus, the antiderivative is:

$\frac{12}{5} x^{5/2} + 16x^{1/2}$

Step 4: Apply the limits of integration

Evaluate the definite integral from $x = 2$ to $x = 8$:

$\left[ \frac{12}{5} x^{5/2} + 16x^{1/2} \right]_2^8$

Evaluate at $x = 8$:

$\frac{12}{5} (8^{5/2}) + 16 (8^{1/2}) = \frac{12}{5} \cdot (64 \sqrt{8}) + 16 \cdot \sqrt{8}$

Evaluate at $x = 2$:

$\frac{12}{5} (2^{5/2}) + 16 (2^{1/2}) = \frac{12}{5} \cdot (4 \sqrt{2}) + 16 \cdot \sqrt{2}$

Step 5: Subtract the results

The definite integral is:

[ \left[ \frac{12}{5} (64 \sqrt{8}) + 16 \sqrt{8} \right] - \left[ \frac{12}{5} (4 \sqrt{2}) + 16 \sqrt{2} \right] ]

If you’d like, I can proceed with a numerical approximation of this expression or leave it in symbolic form. Let me know!

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Would you like a detailed evaluation of the numerical answer, or do you have specific questions about the process?