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Find the Laplace transform?

Find the Laplace transform of f(t)=te^at.

Answer: F(s)=1/(s-a)^2, s>a

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1 Answer

Lf(t) = ∫0 t ete-stdt = ∫te -(s-a)t dt (from 0 to ∞).

Divide and multiply t and dt in the integral by (s-a), introduce new variable z = (s-a)t to come up with the following integral:

                            Lf(t) = (1/(s-a)2) ∫(0,∞) ze-z dz.

You can make sure that the integral (if integrated in parts) equals 1 (-e-z with upper limit z=∞ minus -e-z with z=0). Thus, your answwer is

                                                Lf(t) = 1/(s-a)2