Solve Gaussian integral over finite interval/limits $\int_{a}^{b}xe^{-m(x-t)^2} dx $

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$\int_{a}^{b}xe^{-m(x-t)^2} dx $

where m and t are constants,

ive tried solving this by ...

$\int_{a}^{b}xe^{-m(x-t)^2} dx $ = $\int_{a}^{b}(x-t)e^{-m(x-t)^2} dx $ + t$\int_{a}^{b}e^{-m(x-t)^2} dx $

substituting x-t = u and then

= $\int_{a-t}^{b-t}(u)e^{-m(u)^2} du $ + t$\int_{a-t}^{b-t}e^{-m(u)^2} du $

after that...

as $\int_{a-t}^{b-t}(u)e^{-m(u)^2} du $ = -1/2 $ \int_{a-t}^{b-t}(d/dm)e^{-m(u)^2} du $

not really sure where to go after that or am i on the wrong track???