$\begin{align}  & I=\int{2{{(2x-3)}^{\tfrac{2}{3}}}dx} \\ & \text{Using change of variable method, }u=(2x-3) \\ & \frac{du}{dx}=2,\text{ }dx=\frac{du}{2} \\ & I=\not{2}\int{{{u}^{\tfrac{2}{3}}}\tfrac{du}{{\not{2}}}}=\int{{{u}^{\tfrac{2}{3}}}du} \\ & I=\frac{{{u}^{\tfrac{2}{3}+1}}}{\tfrac{2}{3}+1}=\frac{{{u}^{\tfrac{5}{3}}}}{\tfrac{5}{3}}+C \\ & I=\frac{3}{5}{{u}^{\tfrac{5}{3}}}+C=\frac{3}{5}{{(2x-3)}^{\tfrac{5}{3}}}+C \\\end{align}$