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 \(\boxed{\frac{2}{9}x^{9/2} + 384x^{7/2} + 57344x^{5/2} + \frac{3670016}{3}x^{3/2} + 14x^4 + \frac{17920}{3}x^3 + 344064x^2 + 2097152x}\).