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question about Mathematica integration function

I am having trouble integrating the following functions in

I have a two functions g and g1, where

g = (x/(sigma*Sqrt[2*Pi*t^3]))*Exp[-(x-mu*t)^2/(2*sigma^2*t)]


g1= t*(x/(sigma*Sqrt[2*Pi*t^3]))*Exp[-(x-mu*t)^2/(2*sigma^2*t)]

I assign values for sigma and mu, and then integrate g and g1 with the
limits of integration =BD and 1; I am trying to get an answer in terms
of x. If I use the Integrate function, Mathematica does not solve this
integral, it just restates the integral. If I assign a value for x, it
can numerically solve the integrals if I use NIntegrate, but this is
not very useful to me. In other words, I am trying to 'analytically'
integrate this function.

I realize that I have asked a variation of this question earlier in
the forum, but I feel that if I explicate just what I am interested
in, it may be helpful.

Any advice on how to get Mathematica (or any other software) to
generate the answer?

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