Use a change of variables to evaluate the following indefinite integral. \( \int \frac{(\sqrt{x}+2)^{8}}{2 \sqrt{x}} d x \) Determine a change of variables from \( x \) to u. Choose the correct answer below. A. \( u=\frac{1}{2 \sqrt{x}} \) C. \( u=\sqrt{x} \) D. \( u=\sqrt{x}+2)^{8} \) Write the integral in terms of \( u \). \( \int \frac{(\sqrt{x}+2)^{8}}{2 \sqrt{x}} d x=\int(\square) d u \)

To solve the integral \( \int \frac{(\sqrt{x}+2)^{8}}{2 \sqrt{x}} d x \), a good change of variable to simplify the expression would be \( u=\sqrt{x} \) (option C). The reason this choice is effective is that differentiating \( u \) gives us a straightforward relationship with \( x \) and \( d x \). With this change of variable, we have: - \( x = u^2 \) - \( d x = 2u \, d u \) - \( \sqrt{x} = u \) Now we substitute these into the integral: \[ \int \frac{(u+2)^{8}}{2u} \cdot 2u \, d u \] The \( 2u \) in the numerator and the \( 2u \) in the denominator cancel out, leading us to: \[ \int (u + 2)^{8} \, d u \] So the final expression of the integral in terms of \( u \) is: \[ \int \frac{(\sqrt{x}+2)^{8}}{2 \sqrt{x}} d x = \int (u + 2)^{8} \, d u \]