Laplace transform - split

This is my answer for part a) I can seem to work out how to get to the final answer for e^at and also i am confused about how you would get the ranges for these answers.
Sorry if my work is a bit messy its taken a while to get to this point


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You state in your first post that the Laplace transform of a function, f(x), is ∫0∞f(t)e−stdt\displaystyle \int_0^\infty f(t)e^{-st}dt. So why does the work you show not have any integrals?

The Laplace transform of f(x)=e−ax\displaystyle f(x)= e^{-ax} is ∫0∞e−ate−stdt=∫0∞e−(a+s)tdt\displaystyle \int_0^\infty e^{-at}e^{-st}dt= \int_0^\infty e^{-(a+ s)t}dt. Can you integrate that?

The Laplace transform of f(x)=Ax+B\displaystyle f(x)= Ax+ B is ∫0∞(Ax+B)e−stdt\displaystyle \int_0^\infty (Ax+ B)e^{-st}dt. Integrate that using "integration by parts" with u= Ax+ B and dv=e−stdt\displaystyle dv= e^{-st}dt.
 
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