$\int e^{2 x^3+2 \log _{e} x} dx=$

$\frac{1}{3} e^{2 x^3}+C$ $\frac{1}{6} e^{2 x^3}+C$ $\frac{1}{2} e^{2 x^3}+C$ $\frac{1}{12} e^{2 x^3}+C$

$\frac{1}{6} e^{2 x^3}+C$

$I =\int e^{2 x^3+2 \log x} d x$ $=\int e^{2 x^3} e^{2 \log x} d x$ $=\int e^{2 x^3} e^{\log x^2} d x$ $I =\int x^2 e^{2 x^3} d x$ so let $z = 2x^3$ $dz = 6x^2 dx$ $\frac{dz}{6}=x^2dz$ so $I = \int \frac{e^z}{6}dz$ $\frac{e^z}{6}+C$ $\frac{e^{2 x^3}}{6}+C$