To evaluate the integral

$I={\displaystyle \underset{-}{\overset{}{}}}x{e}^{-6x^2}dx$

you may evaluate the improper integrals

$I_1={\displaystyle \underset{0}{\overset{}{}}}x{e}^{-6x^2}dx=$ and

$I_2={\displaystyle \underset{-}{\overset{0}{}}}x{e}^{-6x^2}dx=.$

These integrals are both convergent in this example, so the value of the given integral is