Answer
$$\frac{1}{4}{e^{{x^4}}} + C $$
Work Step by Step
$$\eqalign{
& \int {{x^3}{e^{{x^4}}}} dx \cr
& {\text{Integrate by substitution method}} \cr
& {\text{Let }}u = {x^4},\,\,\,\,du = 4{x^3}dx,\,\,\,\,dx = \frac{{du}}{{4{x^3}}} \cr
& {\text{Then}}{\text{,}} \cr
& \int {{x^3}{e^{{x^4}}}} dx = \int {{x^3}{e^u}} \left( {\frac{{du}}{{4{x^3}}}} \right) \cr
& {\text{Cancel common factor }}x \cr
& = \frac{1}{4}\int {{e^u}} du \cr
& {\text{integrating}} \cr
& = \frac{1}{4}{e^u} + C \cr
& {\text{replacing }}u = {x^4} \cr
& = \frac{1}{4}{e^{{x^4}}} + C \cr} $$
