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[tex]\int e^{t*(b-s)}[/tex] evaluated from 0 to [tex]\infty[/tex]

So I take the integral and get

[tex]\frac{\int e^{t*(b-s)}}{-(b-s)}[/tex] which evaluated from 0 to [tex]\infty[/tex]

gives me 0 - [tex]\frac{1}{-(b-s)}[/tex]

which is 1/(b-s)

The answer should be 1/(s-b). Can anyone help me figure out what I am messing up?