# Using the Laplace Transform Table in the textbook and Laplace

Using the Laplace Transform Table in the textbook and Laplace Transform Properties,find the (unilateral) Laplace Transforms of the following functions:
$$\displaystyle{t}{\cos{{\left(\omega_{{0}}{t}\right)}}}{u}{\left({t}\right)}$$

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kalfswors0m

$$\displaystyle∵{{\cos{\omega}}_{{0}}{t}}={\frac{{{e}^{{{i}\omega_{{0}}{t}}}+{e}^{{-{i}\omega_{{0}}{t}}}}}{{{2}}}}$$
$$\displaystyle{t}\Rightarrow{\frac{{{1}}}{{{s}^{{2}}}}}$$
$$\displaystyle{t}{{\cos{\omega}}_{{0}}{t}}{u}{\left({t}\right)}\Rightarrow{\frac{{{1}}}{{{2}}}}{\left[{\frac{{{1}}}{{{\left({s}-{i}\omega_{{0}}{t}\right)}^{{2}}}}}+{\frac{{{1}}}{{{\left({s}+{i}\omega_{{0}}{t}\right)}^{{2}}}}}\right]}$$
$$\displaystyle\therefore{L}{\left\lbrace{t}{{\cos{\omega}}_{{0}}{t}}{u}{\left({t}\right)}\right\rbrace}\Rightarrow{\left[{\frac{{{s}^{{2}}-{\omega_{{0}}^{{2}}}}}{{{\left({s}^{{2}}+{\omega_{{0}}^{{2}}}\right)}^{{2}}}}}\right]}$$

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