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Question Number 130806 by shaker last updated on 29/Jan/21

Answered by Olaf last updated on 29/Jan/21

$$\mathrm{sh}x=\frac{2}{3}=\frac{e^x-e^{-x}}{2}$$$$e^{2x}-\frac{4}{3}{e}^x-1=0$$$$e^x=\frac{1}{3}(2+\sqrt{13})$$$$e^{-x}=\frac{3}{2+\sqrt{13}}=\frac{1}{3}(2-\sqrt{13})$$$$e^{3x}=\frac{1}{27}{(2+\sqrt{13})}^3=\frac{1}{27}(86+25\sqrt{13})$$$$e^{-3x}=\frac{1}{27}{(2-\sqrt{13})}^3=\frac{1}{27}(86-25\sqrt{13})$$$$\coth \left(3x\right)=\frac{e^{3x}+e^{-3x}}{e^{3x}-e^{-3x}}$$$$\coth \left(3x\right)=\frac{86}{25\sqrt{13}}$$

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