[tex]\text{Let u = } x^{4}[/tex]
[tex]\frac{du}{dx} = 4x^{3}[/tex]
[tex]\frac{1}{2}\frac{du}{dx} = 2x^{3}[/tex]
[tex]\frac{1}{2}du = (2x^{3})dx[/tex]
Substituting this back into the original integral:
[tex]\int 2x^{3}e^{x^{4}}\, dx = \frac{1}{2}\int e^{u}\, du[/tex]
[tex]= \frac{1}{2}e^{u} + C[/tex]
[tex]= \frac{e^{x^{4}}}{2} + C[/tex]
