Question

Find the Laplace transform using the definition of Laplace cos2 4t

Answer 1

we have to find the Laplace transform of Cos? At using the defination of Laplace Lue know that, The Laplace Transform of a function f(+) is Funetion Fls) which is defined by , -20 F(S) = fc+) -st du Here in the given problem f(+) = cos²47 =¢ * 2605244 ܦܐܘܐ . = [Cos 84 +1] - Cos 20+1 Hones, the Laplaci hransform of f(t) is given by (i) som defination as Tollocus; F($)=* + [684 +1] e-** ** + 1 F(s) est cost at + + + 2 Je-sta dt ti) Calculation : (I) ſesta Cost at Let = ( at by parts Jest cos se at stſe-stot -- SE#(Coset) ſe-star Integrating est lesinst) e het a and (cs öt ) = - a sinar ] CossFiest e-st dt (-5) 8+

A e-St Sin et de 01) A Pastico ***** ***** f**- { [ 4(cinco) Se="de]at [again integrating by parts e-st losst Sint 8 Cos&+. at , A -st casst -st Cosst Sinst 64 [:(sinat) = a.cos ot] Jeste eo&t] e-s o-st cost dt S2 9 A = Sin 8t - Casst s? 64 52 А A ſe-ste -st cos stat A + 64A 52 8 52 SinBt - cat] est ^ (1+*)-( sin st - 65 St Jeast (+ [ $ Sint Casst] eest A - A- Sint $2464) 52 Corat Jenst Jeast Casst at = $ [ 1 Sinst- last jest (II) Se-sta dt - e-st [: Seardt eat ] S Ted e-star = o-st }

From (ü) susing the results (I) and (IT ) I) and (I ) I we have, od -SH 8 Sin (0) (SP+69) 52 240)*+(feien sonjem-219 Ź[0 6:0},"]* 2 [0-4+)] [.2°21 elal and i*-0 at t-r] --- Sen (0-5)]++(+) [:Soko) = * į Sca) 5 + 3s Ž [. Since) = 0 and 63(0) =] S S2+64 So, F(s) = < t sica) *}} ut The Laplace transform of cos? at is Ž (32+64 + 5)