Ta có:

$\int{f\left( x \right)dx}=\int{\sqrt[3]{3x+1}dx}=\int{{{\left( 3x+1 \right)}^{\frac{1}{3}}}.\frac{d\left( 3x+1 \right)}{3}}$

$=\frac{1}{3}.\int{{{\left( 3x+1 \right)}^{\frac{1}{3}}}d\left( 3x+1 \right)}=\frac{1}{3}.\frac{{{\left( 3x+1 \right)}^{\frac{4}{3}}}}{\frac{4}{3}}+C$

$\Rightarrow \int{f\left( x \right)dx}=\frac{1}{4}\left( 3x+1 \right)\sqrt[3]{3x+1}+C$

Vậy đáp án cần tìm là C.