Integrales with mimimun value

find Minimum value of integral Where

soit a>0 , par IPP ,
int(0,1/a) (a^3+4x-a^5x^2)e^(ax)dx
= (1/a) ( (4e/a)-a^3) - (1/a) int(0,1/a) (4-2a^5x)e^(ax) dx

int(0,1/a) (2-a^5x)e^(ax) dx
= (1/a) ( (2-a^4)e-2) +a^4 int(0,1/a) e^(ax) dx
= (1/a) ( (2-a^4)e-2) +a^3(e-1)

==> int(0,1/a) (a^3+4x-a^5x^2)e^(ax)dx
=(1/a) ( (4e/a)-a^3) - (2/a²)( (2-a^4)e-2)-2a²(e-1)
= e(4/a²-2(2-a^4)/a²-2a²)-a²+4/a²+2a²
=a²+4/a²

en étudiant cette fonction: inf_(a>0) (a²+4/a²)=4

thanks abdelbaki.attioui