Solve for: integral of (e^x+10e^{-x})/(e^x-10e^{-x)} x

Expression: $\int{ \frac{ {e}^{x}+10{e}^{-x} }{ {e}^{x}-10{e}^{-x} } } \mathrm{d} x$

Use the substitution $t={e}^{x}-10{e}^{-x}$ to transform the integral

$\int{ \frac{ 1 }{ t } } \mathrm{d} t$

Use $\int{ \frac{ 1 }{ x } } \mathrm{d} x=\ln\left({|x|}\right)$ to evaluate the integral

$\ln\left({|t|}\right)$

Substitute back $t={e}^{x}-10{e}^{-x}$

$\ln\left({|{e}^{x}-10{e}^{-x}|}\right)$

Simplify the expression

$\ln\left({|{e}^{x}-\frac{ 10 }{ {e}^{x} }|}\right)$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\ln\left({|{e}^{x}-\frac{ 10 }{ {e}^{x} }|}\right)+C,& C \in ℝ\end{array}$