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Evaluate $$\mathcal{L}\left\{3^{-t}+\cos h(2 t)\right\}$$

$$\mathcal{L}\left\{e^{\ln \overline{3}^{t}}+\frac{1}{2}\left(e^{2 t}+e^{-2 t}\right)\right\} $$

$$\mathcal{L} e^{-t \ln (3)}+\frac{1}{2}\left(\frac{1}{s-2}+\frac{1}{s+2}\right)$$

$$\mathcal{L}=e^{-t \ln (3)}+\frac{1}{2}\left(\frac{1(s+2)}{(s-2)(s+2)}+\frac{1(s-2)}{s+2(s-2)}\right)$$

$$\frac{1}{s+\ln (3)}+\frac{1}{2}\left(\frac{2s}{s^{2}-4}\right)=$$

$$\frac{1}{s+\ln (3)}+\frac{s}{s^{2}-4} $$

Find the Laplace transform of the following function $$\cos h t \sin ^{2} t$$

$$\mathcal{L}\left\{\cos h t \sin ^{2} t\right\}=\mathcal{L}\left\{\frac{e^{t}+e^{t}}{2} \cdot \frac{1}{2}(1-\cos (2 t))\right\}$$

$$\frac{1}{4} \mathcal {L}\left\{\left(e^{t}+e^{-t}\right)(1-\cos (2 t))\right\} $$

$$\frac{1}{4} \mathcal {L}\left\{e^{t}-e^{t} \cos (2 t)+e^{-t}-e^{-t} \cos (2 t)\right\} $$

$$\frac{1}{4}\left[\frac{1}{s-1}-\left(\frac{s}{s^{2}+4}\right)+\frac{1}{s+1}-\left (\frac{s}{s^{2}+4}\right)\right].$$

$$\frac{1}{4}\left[\frac{1}{s-1}-\frac{s-1}{(s-1)^{2}+4}+\frac{1}{s+1}-\frac{s-1}{(s-1)^{2}+4}\right]$$

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