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2017-05-22 16:48:13 +0100 answered a question Abstract symbolic matrix exponential
t, s = var(' t s ')

A = matrix([[3*t,t^2],[-2*t, t]])

B = A.apply_map(lambda e: integrate(e,t,0,s))

B.exp()

I Think this is what you are looking for. Note that simple 'integrate' function won't integrate the matrix.