Let ℒ{𝑓(𝑡)} = 𝐹(𝑠). Show that 𝑓(𝑡) = − 1 𝑡 ℒ −1 {𝐹 ′ (𝑠)} Thus, if we know how to invert 𝐹 ′ (𝑠) then we know how to invert 𝐹(𝑠). Use this information to find the Laplace inverse transform of (i) arctan ( 𝑎 𝑠 ), (ii) ln ( 𝑠+𝑎 𝑠−𝑎 ).
"F(s)=\\int^{\\infin}_0e^{-ts}f(t)dt"
"F'(s)=e^{-ts}f(t)"
"L^{-1}(F(s))=f(t)"
"L^{-1}(F'(s))=f(t)\/(-t)"
"f(t)=-tL^{-1}(F'(s))"
i)
"F'(s)=\\frac{a}{1+s^2}"
"f(t)=-atsint"
ii)
"F'(s)=\\frac{1}{s+a}-\\frac{1}{s-a}"
"f(t)=-t(e^{-at}-e^{at})"
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