How do you integrate x* (exp(x))??

The easiest method to use in this incidence is integratation by parts.

So let u=x and dv/dx=exp(x). Therefore du/dx=1 and v=exp(x).

Then we use the formula where integral(udu/dx)=uv-integral(v*du/dx).

So integral(xexp(x))=xexp(x)-integral(exp(x)*1)

=x*exp(x)-integral(exp(x))

=x*exp(x)-exp(x)+c

Don't forget the +c