∫ e^2 x^3+2 log _e x dx=

$I =\int e^{2 x^3+2 \log x} d x$

$=\int e^{2 x^3} e^{2 \log x} d x$

$=\int e^{2 x^3} e^{\log x^2} d x$

$I =\int x^2 e^{2 x^3} d x$

so let $z = 2x^3$

$dz = 6x^2 dx$

$\frac{dz}{6}=x^2dz$

so $I = \int \frac{e^z}{6}dz$

$\frac{e^z}{6}+C$

$\frac{e^{2 x^3}}{6}+C$