$12.56 \int_{0}^{7 / 3} \frac{x+1}{\sqrt[3]{3 x+1}} d x$.

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$12.56 \int_{0}^{7 / 3} \frac{x+1}{\sqrt[3]{3 x+1}} d x$.
12.56 We have

\[
\begin{aligned}
& \int_{0}^{7 / 3} \frac{x+1}{\sqrt[3]{3 x+1}} d x=\int_{0}^{7 / 3} \frac{3(x+1)}{3 \sqrt[3]{3 x+1}} d x=\frac{1}{3} \int_{0}^{7 / 3} \frac{3 x+1}{\sqrt[3]{3 x+1}} d x+ \\
& +\frac{1}{3} \int_{0}^{7 / 3} \frac{2}{\sqrt[3]{3 x+1}} d x=\frac{1}{3}\left(\int_{0}^{7 / 3}(3 x+1)^{2 / 3} d x+2 \int_{0}^{7 / 3}(3 x+1)^{-1 / 3} d x\right)= \\
& =\frac{1}{3}\left(\left.\frac{1}{3} \frac{(3 x+1)^{5 / 3}}{\frac{5}{3}}\right|_{0} ^{7 / 3}+\left.\frac{2}{3} \frac{(3 x+1)^{2 / 3}}{\frac{2}{3}}\right|_{0} ^{7 / 3}\right)= \\
& =\frac{1}{3}\left(\frac{1}{5}\left(8^{5 / 3}-1\right)+\left(8^{2 / 3}-1\right)\right)=\frac{46}{15}
\end{aligned}
\]

Answer: $\frac{46}{15}$.