6. Use the table below given $\int_{0}^{1} f^{\prime}(x) g(x) d x=7$, then find $\int_{0}^{1} f(x) g^{\prime}(x) d x=$ \begin{tabular}{|c|c|c|} \hline$x$ & 0 & 1 \\ \hline$f(x)$ & 3 & 5 \\ \hline$f^{\prime}(x)$ & -1 & -5 \\ \hline$g(x)$ & -3 & 4 \\ \hline$g \prime(x)$ & 0 & 1 \\ \hline \end{tabular}

Step 1 ：Given the integral \(\int_{0}^{1} f^{\prime}(x) g(x) d x=7\), we need to find the value of \(\int_{0}^{1} f(x) g^{\prime}(x) d x\).

Step 2 ：We can use the integration by parts formula, which states that \(\int u dv = uv - \int v du\). In this case, we can let \(u = f(x)\) and \(v = g(x)\).

Step 3 ：According to the formula, \(\int_{0}^{1} f(x) g^{\prime}(x) d x = f(x)g(x) - \int_{0}^{1} f^{\prime}(x) g(x) d x\), evaluated from 0 to 1.

Step 4 ：Substituting the given values into this expression, we have \(f(1)g(1) - f(0)g(0) - \int_{0}^{1} f^{\prime}(x) g(x) d x = 5*4 - 3*(-3) - 7 = 20 + 9 - 7 = 22\).

Step 5 ：So, the value of \(\int_{0}^{1} f(x) g^{\prime}(x) d x\) is \(\boxed{22}\).