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23/09/2017

As primitivas da secante hiperbólica

Exercício:
Determine uma expressão para \[ \int {\sech x} dx\]
\[\arcsen \left(\th x\right) +C_1\] ou \[2\arctg{\left(e^x\right)}+C_2 \] ou \[\arctg{\left(\sh x\right)}+C_3 \] ou \[2\arctg \left[\th \left(\displaystyle\frac{x}{2}\right)\right] + C_4\]

\begin{eqnarray*} \int {\sech x} dx & = & \int {\displaystyle\frac{\sechq x}{\sech x}} dx \\ & = & \int {\displaystyle\frac{ (\th x)' }{\sqrt{\sechq x} } }dx \\ & = & \int {\displaystyle\frac{ (\th x)' }{\sqrt{1- \thEL{2} x} } }dx\\ & = & \arcsen \left(\th x\right) + C_1 \end{eqnarray*}


\begin{eqnarray*} \int {\sech x} dx & = & \int {\displaystyle\frac{2}{e^x+e^{-x}}} dx \\ & = & \int {\displaystyle\frac{ 2e^x }{\left(e^x\right)^2+1 } }dx \\ & = & 2\arctg{\left(e^x\right)}+C_2 \end{eqnarray*}
Ver http://ftp.ist.utl.pt/GAEL/math/integrals/more/sech.htm


\begin{eqnarray*} \int {\sech x} dx & = & \int {\displaystyle\frac{1}{\ch x} dx} \\ & = & \int {\displaystyle\frac{\ch x }{\chEL{2}x }dx} \\ & = & \int {\displaystyle\frac{\ch x }{\shEL{2}x+1 }dx}\\ & = & \arctg\left(\sh x\right)+C_3 \end{eqnarray*}
Ver http://ftp.ist.utl.pt/GAEL/math/integrals/tableof.htm


Pode-se proceder de forma análoga ao que se fez para a secante trigonométrica
Calculemos as primitivas da função secante hiperbólica, recorrendo à substituição \[t = \th \left(\displaystyle\frac{x}{2}\right)\]. Recorde-se que \[ \ch x = \displaystyle\frac{{1 + t^2 }}{{1 - t^2 }} \] e \[ \sh x = \displaystyle\frac{{2t}}{{1 - t^2 }} \] E portanto \begin{eqnarray*} \int {\sech x} dx & = & \int {\displaystyle\frac{1}{\ch x}} dx \\ & = & \int {\displaystyle\frac{1 - t^2 }{1 + t^2 } \times \displaystyle\frac{2}{1 - t^2 }}dt \\ & = & \int{\displaystyle\frac{2}{1 + t^2} } dt\\ & = & 2\arctg t + C_4\\ & = & 2\arctg \left[\th \left(\displaystyle\frac{x}{2}\right)\right] + C_4 \end{eqnarray*} Resultado curioso...
Note-se que \[\left(\sech x\right)'=-\th x \sech x\] e \[\left(\th x\right)'=\sechq x\]. Portanto, multiplicar e dividir por $\th x + \sech x$ não nos conduz a algo como o simpático resultado que temos para a secante trigonométrica...

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