Question

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 ). 

Answer 1

$F(s)=\int^{\infin}_0e^{-ts}f(t)dt$

$F'(s)=e^{-ts}f(t)$

$L^{-1}(F(s))=f(t)$

$L^{-1}(F'(s))=f(t)/(-t)$

$f(t)=-tL^{-1}(F'(s))$

i)

$F'(s)=\frac{a}{1+s^2}$

$f(t)=-atsint$

ii)

$F'(s)=\frac{1}{s+a}-\frac{1}{s-a}$

$f(t)=-t(e^{-at}-e^{at})$