Ejemplo 8
Encontrar la {{\mathfrak{F}}_{c}}\left[ {{e}^{-at}} \right] y la {{\mathfrak{F}}_{s}}\left[ {{e}^{-at}} \right].
{{\mathfrak{F}}_{c}}\left[ {{e}^{-at}} \right]=\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t\text{ }\!\!~\!\!\text{ }dt
Se definen: u=Cos\text{ }\!\!~\!\!\text{ }\omega t,\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }du=-\omega .Sen\omega tdt,\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }dv={{e}^{-at}}\text{ }\!\!~\!\!\text{ }dt,\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }v=\frac{-{{e}^{-at}}}{a}
\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t\text{ }\!\!~\!\!\text{ }dt=\left. \frac{-{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t}{a} \right|\begin{matrix}
\infty \\
0 \\
\end{matrix}-\frac{\omega }{a}\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Sen\omega tdt
Se definen: u=Sen\text{ }\!\!~\!\!\text{ }\omega t,\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }du=\omega .Cos\text{ }\!\!~\!\!\text{ }\omega tdt,\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }dv={{e}^{-at}}\text{ }\!\!~\!\!\text{ }dt,\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }v=\frac{-{{e}^{-at}}}{a} para la segunda integral.
\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t\text{ }\!\!~\!\!\text{ }dt=\frac{1}{a}-\frac{\omega }{a}\left[ \left. \frac{-{{e}^{-at}}Sen\text{ }\!\!~\!\!\text{ }\omega t}{a} \right|\begin{matrix}
\infty \\
0 \\
\end{matrix}+\frac{\omega }{a}\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega tdt \right]
\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t\text{ }\!\!~\!\!\text{ }dt=\frac{1}{a}-\frac{{{\omega }^{2}}}{{{a}^{2}}}\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega tdt
\left( 1+\frac{{{\omega }^{2}}}{{{a}^{2}}} \right)\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t\text{ }\!\!~\!\!\text{ }dt=\frac{1}{a}
\underset{0}{\overset{\infty }{\mathop \int }}\,{{e}^{-at}}Cos\text{ }\!\!~\!\!\text{ }\omega t\text{ }\!\!~\!\!\text{ }dt=\frac{1}{a}\left( \frac{{{a}^{2}}}{{{\omega }^{2}}+{{a}^{2}}} \right)
Por lo tanto, {{\mathfrak{F}}_{c}}\left[ {{e}^{-at}} \right]=\frac{a}{{{\omega }^{2}}+{{a}^{2}}}.
Análogamente, {{\mathfrak{F}}_{s}}\left[ {{e}^{-at}} \right]=\frac{\omega }{{{\omega }^{2}}+{{a}^{2}}}.
