If $f\left(x\right)=\int e^{x^2}(x-2)(x-3)(x-4)dx$

then $f$ increases on

$f^{\mathrm{\prime }}\left(x\right)=e^{x^2}(x-2)(x-3)(x-4)$

As $e^{x^2}>0$ for all $x\in \mathbf{R},f^{\mathrm{\prime }}\left(x\right)>0$

if $(x-2) (x-3) (x-4)>0$ i.e., if $2<x<3\text{ or }x>4$

Thus $f\left(x\right)$ increases on the interval$(2, 3)$