Use a change of variables to evaluate the following indefinite integral.
\[
\int \frac{(\sqrt{x}+8)^{7}}{2 \sqrt{x}} d x
\]
Determine a change of variables from $\times$ to $\mathrm{u}$. Choose the correct answer below.
A. $u=(\sqrt{x}+8)^{7}$
B. $u=\sqrt{x}+8$
c. $u=\sqrt{x}$
$\mathrm{u}=\frac{1}{2 \sqrt{x}}$
Write the integral in terms of $u$.
\[
\int \frac{(\sqrt{x}+8)^{7}}{2 \sqrt{x}} d x=\int(\square) d u
\]
Evaluate the integral
\[
\int \frac{(\sqrt{x}+8)^{7}}{2 \sqrt{x}} d x=
\]

Step 1 ：Choose the correct change of variables from x to u. The correct choice is \(u = \sqrt{x} + 8\).

Step 2 ：Write the integral in terms of u. The integral \(\int \frac{(\sqrt{x}+8)^{7}}{2 \sqrt{x}} dx\) becomes \(\int u^7 du\) after the substitution.

Step 3 ：Evaluate the integral \(\int u^7 du\). The result is a polynomial of degree 9/2.

Step 4 ：The integral of \(\int u^7 du\) is \(\frac{2}{9}x^{9/2} + 384x^{7/2} + 57344x^{5/2} + \frac{3670016}{3}x^{3/2} + 14x^4 + \frac{17920}{3}x^3 + 344064x^2 + 2097152x\).

Step 5 ：Final Answer: The integral of \(\int \frac{(\sqrt{x}+8)^{7}}{2 \sqrt{x}} dx\) is \(\boxed{\frac{2}{9}x^{9/2} + 384x^{7/2} + 57344x^{5/2} + \frac{3670016}{3}x^{3/2} + 14x^4 + \frac{17920}{3}x^3 + 344064x^2 + 2097152x}\).