\( \displaystyle \int (e^{2\ln x}+e^{2x}) dx = \)
\(2+\dfrac{e^{2x}}{2}+C\)
\( \dfrac{e^{x^3}}{3}+2e^{2x}+C \)
\( \dfrac{e^{x^3}}{3}+\dfrac{e^{2x}}{2}+C \)
\(\dfrac{x^3}{3}+\dfrac{e^{2x}}{2}+C \)
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$$ \int (e^{2\ln x}+e^{2x}) dx = \int (e^{\ln x^2}+e^{2x}) dx $$ $$ = \int (x^2+e^{2x}) dx $$ $$ = \boxed{\frac{x^3}{3} + \frac{e^{2x}}{2} + C} $$