Question

Evaluate the integral

Answer 1

$\text{Let u = } x^{4}$
$\frac{du}{dx} = 4x^{3}$
$\frac{1}{2}\frac{du}{dx} = 2x^{3}$
$\frac{1}{2}du = (2x^{3})dx$

Substituting this back into the original integral:
$\int 2x^{3}e^{x^{4}}\, dx = \frac{1}{2}\int e^{u}\, du$
$= \frac{1}{2}e^{u} + C$

$= \frac{e^{x^{4}}}{2} + C$