Integrate x*e^(-x^2)dx?

Question

Q.Integrate x*e^(-x^2) dx
Can anyone see how to do this?
I can't get the product rule to work because I can't integrate the second bit.
Thanks

Answer 1

Use substitution

u = -x^2
du = -2x dx

Ana

Answer 2

Well, the derivative of e^u = du*e^u, so we'll need to get the formula into that format before integrating.
if u = -(x^2), then du = -2x,
now if
solution V = Integral( x * e^(-(x^2)) ), but we need to the first part to be -2x, not x, so we multiply both sides by -2.

-2 * V = Int ( -2x * e^(-(x^2) ) )... now we have something we can integrate on the right side.

-2* V = e^(-(x^2) ) + C... now solve for V by dividing by -2 again.
V = - (e^(-x^2) / 2) + C (note, since C is arbitrary, C and C/2 are the same thing, so we can just pull it out of the combined fraction).

ok?

Answer 3

Integrate x*e^(-x^2)dx
= Integrate e^(-x^2) (-1/2) d-x^2, mental substitution
= (-1/2)e^(-x^2)+c