definite integration

can someone find the end result of the definite integral of xe^(-2x), from 1 to infinity?

i did it on a recent exam, and i secured the answer 3/4e^2.  although the answer was/is 'wrong', i got the question right since it was checking for convergence or divergence(either way, it would've ended up converging).  my professor said that the true answer was 1/4e^2.  i'm checking over my exam, and i don't see where i went wrong with the signs unless i'm missing something? i used classic FTC, f(b) - f(a), and i still don't know how he was able to subtract 1 from 2.

fyi, this is calc II material.  the class is calc III, but he adds in series-questions every now and then.  can someone help me out?

>>6
Technically, Riemann integration is defined only on bounded intervals, no?

Is an improper integral like this still said to be "Riemann integrable"? Or is it technically only "Lebesgue integrable", but just able to be computed by Riemann means (and taking the limit of the endpoint of the interval)?

I'm just trying to see what the point of Lebesgue integration is...