Let's evaluate the given definite integral:

$\int_3^8 \frac{6x^2 + 8}{\sqrt{x}} \, dx$

Step 1: Simplify the integrand

We can rewrite the integrand by splitting it into two terms and simplifying:

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

Thus, the integral becomes:

$\int_3^8 \left( 6x^{\frac{3}{2}} + 8x^{-\frac{1}{2}} \right) dx$

Step 2: Integrate term by term

We use the power rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1}$, for each term:

1. For $6x^{\frac{3}{2}}$: $\int 6x^{\frac{3}{2}} \, dx = 6 \cdot \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{12}{5} x^{\frac{5}{2}}$

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

The indefinite integral is: $\frac{12}{5} x^{\frac{5}{2}} + 16x^{\frac{1}{2}}$

Step 3: Apply limits of integration

Now we evaluate the definite integral from $x = 3$ to $x = 8$. Substituting the limits into the antiderivative:

$\int_3^8 \frac{6x^2 + 8}{\sqrt{x}} \, dx = \left[ \frac{12}{5} x^{\frac{5}{2}} + 16x^{\frac{1}{2}} \right]_3^8$

Evaluate at $x = 8$:

1. $x^{\frac{5}{2}} = 8^{\frac{5}{2}} = (8^{\frac{1}{2}})^5 = 2^5 = 32$, so $\frac{12}{5} x^{\frac{5}{2}} = \frac{12}{5} \cdot 32 = \frac{384}{5}$.
2. $x^{\frac{1}{2}} = 8^{\frac{1}{2}} = 2$, so $16x^{\frac{1}{2}} = 16 \cdot 2 = 32$.

Adding these: $\frac{12}{5} \cdot 32 + 32 = \frac{384}{5} + \frac{160}{5} = \frac{544}{5}$

Evaluate at $x = 3$:

1. $x^{\frac{5}{2}} = 3^{\frac{5}{2}} = (3^{\frac{1}{2}})^5 = (\sqrt{3})^5$, so $\frac{12}{5} x^{\frac{5}{2}} = \frac{12}{5} \cdot (\sqrt{3})^5$.
2. $x^{\frac{1}{2}} = \sqrt{3}$, so $16x^{\frac{1}{2}} = 16 \cdot \sqrt{3}$.

Adding these: $\frac{12}{5} \cdot (\sqrt{3})^5 + 16\sqrt{3}$

Step 4: Final result

The definite integral is:

$\int_3^8 \frac{6x^2 + 8}{\sqrt{x}} \, dx = \frac{544}{5} - \left( \frac{12}{5} \cdot (\sqrt{3})^5 + 16\sqrt{3} \right)$

Would you like the numeric approximation for this result or additional details about any step?

Related Questions:

1. What are the steps to convert a fractional exponent to radicals?
2. How does the power rule of integration handle negative exponents?
3. Why do we split the integrand before integration?
4. How do definite integrals differ from indefinite integrals?
5. What are practical applications of integrals with fractional exponents?

Tip:

For definite integrals, always simplify the integrand first to make integration straightforward.