Important Laplace Transforms

\displaystyle f{{\left({t}\right)}}\xrightarrow{\text{LT}}{\int_{ {0}}^{\infty}} f{{\left({t}\right)}}{e}^{ -{{st}}}{\left.{d}{t}\right.}

  1. \displaystyle\delta{\left({t}\right)}\xrightarrow{\text{LT}}{1}

  2. \displaystyle{u}{\left({t}\right)}\xrightarrow{\text{LT}}\frac{1}{{s}}

  3. \displaystyle{t}\xrightarrow{\text{LT}}\frac{1}{{s}^{2}}

  4. \displaystyle{t}^{n}{\left({n}>{0}\right)}\xrightarrow{\text{LT}}\frac{n!}{{s}^{{{n}+{1}}}}

  5. \displaystyle{e}^{ -{{at}}}\xrightarrow{\text{LT}}\frac{1}{{s + a}}

  6. \displaystyle \sin{{b}}{t}\xrightarrow{\text{LT}}\frac{b}{{s}^{2}+{b}^{2}}

  7. \displaystyle \cos{{b}}{t}\xrightarrow{\text{LT}}\frac{s}{{s}^{2}+{b}^{2}}

  8. \displaystyle{e}^{ -{{at}}} \sin{\omega}{t}\xrightarrow{\text{LT}}\frac{\omega}{{\left({s}+{a}\right)}^{2}+{w}^{2}}

  9. \displaystyle{e}^{ -{{at}}} \cos{\omega}{t}\xrightarrow{\text{LT}}\frac{s+a}{{\left({s}+{a}\right)}^{2}+{w}^{2}}

  10. \displaystyle\frac{ df{{\left({t}\right)}}}{{\left.{d}{t}\right.}}\xrightarrow{\text{LT}}{s}{F}{\left({s}\right)}- f{{\left({0}\right)}}

  11. \displaystyle\int f{{\left({t}\right)}}{\left.{d}{t}\right.}\xrightarrow{\text{LT}}\frac{1}{{s}}{F}{\left({s}\right)}

  12. \displaystyle f{{\left({t}-{a}\right)}}\xrightarrow{\text{LT}}{e}^{ -{{as}}}{F}{\left({s}\right)}

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