What the integral of e^2x*x? (limits 0,1)

Whenever you see an integral with a product as the integrand, check if integration by parts works.
That is,
Int(v(x)*u'(x)) = u(x)v(x)-int(u(x)v'(x))
So you should see that its easier to call v(x)=x and u'(x)=e^2x
So then we get v'(x)=1 and u(x)=1/2e^2x
So therefore using the by parts formula the integral goes to
1/2xe^2x-int(1/2e^2x) = 1/2xe^2x -1/4e^2x
when you apply the limits you get (e^2+1)/4