Complete the table by identifying $u$ and $d u$ for the integral.
\begin{tabular}{|l|c|c|}
\hline $\int f(g(x)) g^{\prime}(x) d x$ & $u=g(x)$ & $d u=g^{\prime}(x) d x$ \\
\hline $\int x^{7} \sqrt{x^{8}+3} d x$ & $u=\square d u=\square$ \\
\hline
\end{tabular}

Step 1 ：The integral is in the form of \( \int f(g(x)) g^{\prime}(x) d x \). Here, \( f(g(x)) \) is \( x^{7} \sqrt{x^{8}+3} \) and \( g^{\prime}(x) \) is the derivative of some function \( g(x) \). We need to identify \( g(x) \) such that its derivative is a part of the integrand.

Step 2 ：Looking at the integrand, we can see that \( x^{7} \) is the derivative of \( \frac{1}{8}x^{8} \). Also, \( \sqrt{x^{8}+3} \) can be written as \( (x^{8}+3)^{1/2} \). So, if we let \( u=x^{8}+3 \), then \( du=8x^{7}dx \).

Step 3 ：However, in the integral, we have \( x^{7}dx \) not \( 8x^{7}dx \). So, we need to adjust the \( du \) by dividing it by 8. Therefore, \( du=\frac{1}{8}8x^{7}dx=x^{7}dx \).

Step 4 ：So, \( u=x^{8}+3 \) and \( du=x^{7}dx \).

Step 5 ：Final Answer: For the integral \( \int x^{7} \sqrt{x^{8}+3} d x \), \( u=x^{8}+3 \) and \( du=x^{7}dx \). So, the completed table is: \n \begin{tabular}{|l|c|c|} \n \hline \( \int f(g(x)) g^{\prime}(x) d x \) & \( u=g(x) \) & \( d u=g^{\prime}(x) d x \) \ \n \hline \( \int x^{7} \sqrt{x^{8}+3} d x \) & \( u=x^{8}+3 \) & \( d u=x^{7}dx \) \ \n \hline \n \end{tabular}