L-means-laplacr-trsnsform-find-L-sin-at-and-L-cos-at-

Question Number 28993 by abdo imad last updated on 02/Feb/18

L m e a n s l a p l a c r t r s n s f o r m f i n d L (s i n (a t))

a n d L (c o s (a t)) .

Answered by sma3l2996 last updated on 03/Feb/18

L (s i n (a t)) = \int_{0}^{\infty} s i n (a t) e^{- s t} d t

u = s i n (a t) \Rightarrow u^{'} = a c o s (a t)

v^{'} = e^{- s t} \Rightarrow v = - \frac{1}{s} e^{- s t}

L (s i n (a t)) = \frac{a}{s} \int_{0}^{\infty} c o s (a t) e^{- s t} d t

u = c o s (a t) \Rightarrow u^{'} = - a s i n (a t)

v^{'} = e^{- s t} \Rightarrow v = - \frac{1}{s} e^{- s t}

L (s i n (a t)) = \frac{a}{s^{2}} - \frac{a^{2}}{s^{2}} \int_{0}^{\infty} s i n (a t) e^{- s t} d t = \frac{a}{s^{2}} - \frac{a^{2}}{s^{2}} L (s i n (a t))

L (s i n (a t)) (1 + \frac{a^{2}}{s^{2}}) = \frac{a}{s^{2}}

L (s i n (a t)) = \frac{a}{s^{2} + a^{2}}