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Consider the following.
\[
\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} d x
\]
Use algebra to rewrite the integrand using rational exponents.
$\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} d x=\int_{1}^{9}\left(8 x^{3 / 2}+x^{5 / 2}\right) d x$
$\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} d x=\int_{1}^{9}\left(8 x^{1 / 2}+x^{-1 / 2}\right) d x$
$\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} d x=\int_{1}^{9}\left(8 x^{1 / 2}+x^{3 / 2}\right) d x$
$\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} d x=\int_{1}^{9}\left(8 x^{1 / 2}+x^{5 / 2}\right) d x$
$\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} d x=\int_{1}^{9}\left(8 x^{-1 / 2}+x^{3 / 2}\right) d x$
Evaluate the integral.

Step 1 ：Consider the integral \(\int_{1}^{9} \frac{8+x^{2}}{\sqrt{x}} dx\)

Step 2 ：Use algebra to rewrite the integrand using rational exponents. This can be done by splitting the fraction into two terms and rewriting \(\sqrt{x}\) as \(x^{1/2}\). This gives us \(\int_{1}^{9}(8x^{1/2} + x^{3/2}) dx\)

Step 3 ：The integral can be rewritten as the sum of two separate integrals, each of which can be solved using the power rule for integration. The power rule states that the integral of \(x^n dx\) is \((1/(n+1))x^{(n+1)}\)

Step 4 ：Applying the power rule to each term in the integral gives us \(\frac{16}{3}x^{3/2} + \frac{2}{5}x^{5/2}\)

Step 5 ：Evaluating this from 1 to 9 gives us \(\frac{16}{3}(9^{3/2} - 1^{3/2}) + \frac{2}{5}(9^{5/2} - 1^{5/2})\)

Step 6 ：Simplifying this gives us \(\frac{644}{5}\)

Step 7 ：Final Answer: The integral of the function from 1 to 9 is \(\boxed{\frac{644}{5}}\)