6.4. Using the transform pairs in Table 6.2 and the properties of the Laplace transform in Table 6.1, determine the Laplace transform of the following signals: (a) $x(t) = (e^{-bt} \cos^2 \omega t)u(t)$ (b) $x(t) = (e^{-bt} \sin^2 \omega t)u(t)$ (c) $x(t) = (t \cos^2 \omega t)u(t)$
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Using the transform pair in Table 6.2, we have: L{e^(-b*cos(wt)*u(t))} = 1 / (s + b*cos(wt)) Show more…
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6.10 Using the transform pairs in Table 6.1 and the properties of the Laplace transform, prove the following Laplace transform pairs:
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6.29 Use the basic Laplace transforms and the Laplace transform properties given in Tables D.1 and D.2 to determine the unilateral Laplace transform of the following signals: (a) x(t) = d/dt {te^-t u(t)} (b) x(t) = tu(t) * cos(2̴̵̶̷̸̡̢̧̨̛̖̗̘̙̜̝̞̟̠̣̤̥̦̩̪̫̬̭̮̯̰̱̲̳̹̺̻̼͇͈͉͍͎̀́̂̃̄̅̆̇̈̉̊̋̌̍̎̏̐̑̒̓̔̽̾̿̀́͂̓̈́͆͊͋͌̕̚ͅ͏͓͔͕͖͙͚͐͑͒͗͛ͣͤͥͦͧͨͩͪͫͬͭͮͯ͘͜͟͢͝͞͠͡ͰͱͲͳʹ͵Ͷͷͺͻͼͽ;Ϳ΄΅Ά·ΈΉΊΌΎΏΐΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩΪΫάέήίΰαβγδεζηθικλμνξοπt)u(t) (c) x(t) = t^3 u(t) (d) x(t) = u(t - 1) * e^-2t u(t - 1) (e) x(t) = ∫[0 to t] e^-3τ cos(2τ) dτ (f) x(t) = t d/dt (e^-t cos(t) u(t))
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Determine the Laplace transform of each of the following functions: (a) $f_{1}(t)=\left\{\begin{array}{ll}{t,} & {t=1,2,3, \ldots,} \\ {e^{t},} & {t \neq 1,2,3, \ldots.}\end{array}\right.$ (b) $f_{2}(t)=\left\{\begin{array}{ll}{e^{t},} & {t \neq 5,8,} \\ {6,} & {t=5}, \\ {0,} & {t=8.}\end{array}\right.$
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