calculate-L-e-2x-cos-pix-L-laplace-transform-

Question Number 96660 by mathmax by abdo last updated on 03/Jun/20

calculate L (e^{- 2 x} \cos (π x)) L laplace transform

Answered by mathmax by abdo last updated on 04/Jun/20

L (e^{- 2 x} \cos (π x)) = \int_{0}^{\infty} e^{- 2 t} \cos (π t) e^{- xt} dt

= \int_{0}^{\infty} e^{- (x + 2) t} \cos (π t) dt = Re (\int_{0}^{\infty} e^{- (x + 2) t} e^{i π t} dt)

= Re (\int_{0}^{\infty} e^{(- x - 2 + i π) t} dt) we have

\int_{0}^{\infty} e^{(- x - 2 + i π) t} dt = {[\frac{1}{- x - 2 + i π} e^{(- x - 2 + i π) t}]}_{0}^{+ \infty}

= - \frac{1}{- x - 2 + i π} = \frac{1}{x + 2 - i π} = \frac{x + 2 + i π}{{(x + 2)}^{2} + π^{2}} \Rightarrow

L (e^{- 2 x} \cos (π x)) = \frac{x + 2}{{(x + 2)}^{2} + π^{2}}