f-t-1-cos2t-t-find-laplace-transformation-

Question Number 128485 by BHOOPENDRA last updated on 07/Jan/21

f (t) = \frac{1 - c o s 2 t}{t}

f i n d l a p l a c e t r a n s f o r m a t i o n ?

Answered by Dwaipayan Shikari last updated on 07/Jan/21

L (\frac{1 - c o s 2 t}{t}) \Rightarrow I (s) = \int_{0}^{\infty} e^{- s t} \frac{1 - c o s 2 t}{t} d t

I^{'} (s) = - \int_{0}^{\infty} e^{- s t} d t + \int_{0}^{\infty} e^{- s t} c o s 2 t = - \frac{1}{s} + \frac{1}{2} \int_{0}^{\infty} e^{- s t + 2 i t} + e^{- s t - 2 i t}

= - \frac{1}{s} + \frac{1}{2} (\frac{1}{s - 2 i} + \frac{1}{s + 2 i}) = - \frac{1}{s} + \frac{s}{s^{2} + 4}

I (s) = l o g (\frac{\sqrt{s^{2} + 4}}{s}) + C \Rightarrow I (s) = l o g (\frac{\sqrt{s^{2} + 4}}{s}) (C = 0)

Commented by BHOOPENDRA last updated on 07/Jan/21

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