Numerical integral cannot be plotted

I am trying to plot M[x]=e^(x^2) and the integral solution of I= ∫x^2*M(x) dx on the same plot.
I did the following:

F[x_] := e^(x^2)
Lamb = Plot[F[x], {x, 0, 10}] to get the one of the graphs.

Now to plot the integral solution I did this:

MI[x_] := x^2*F[x]
Integrate[MI[x], {x, 0, 20}]
46.3718

To plot it I did the following:

u[x_?NumericQ] := NIntegrate[(x^2*E^x^2), {x, 0, 2}]
Plot[u[x], {x, 0, 20}]

However, I am not getting a graph for the integral solution. Is it possible to get one? If it is, I want to plot F(x) and the integral solution on the same plot.

Clear[u, x]
u[z_?NumericQ] = Integrate[(x^2*E^x^2), {x, 0, z}];
LogPlot[u[x], {x, 0, 20}]