My teacher gave us a worksheet and sometimes I see csch(...) and tanh(...) and other times without the "h" Does the h mean anything or is this probably a typo?
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They're the Hyperbolic Functions, we define as follows $\displaystyle \sinh=\frac{e^x-e^{-x}}2$ and $\displaystyle \cosh=\frac{e^x+e^{-x}}2$
Originally Posted by Krizalid They're the Hyperbolic Functions, we define as follows $\displaystyle \sinh=\frac{e^x-e^{-x}}2$ and $\displaystyle \cosh=\frac{e^x+e^{-x}}2$ based on that... is tanh = $\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}} $ ?
Originally Posted by circuscircus based on that... is tanh = $\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}} $ ? yes, $\displaystyle \tanh x = \frac {e^x - e^{-x}}{e^x + e^{-x}}$ but it is usually just defined as $\displaystyle \frac {\sinh x}{\cosh x}$
Originally Posted by Jhevon yes, $\displaystyle \tanh x = \frac {e^x - e^{-x}}{e^x + e^{-x}}$ but it is usually just defined as $\displaystyle \frac {\sinh x}{\cosh x}$ how do you integrate these functions? sinh, cosh and tanh?
Originally Posted by circuscircus how do you integrate these functions? sinh, cosh and tanh? $\displaystyle \int \sinh x dx = \cosh x + C$ $\displaystyle \int \cosh x dx = \sinh x +C$ $\displaystyle \int \tanh x dx = \int \frac{e^x - e^{-x}}{e^x + e^{-x}} dx$ let $\displaystyle u=e^{x}+e^{-x}$. Continue.
Originally Posted by circuscircus how do you integrate these functions? sinh, cosh and tanh? $\displaystyle \begin{aligned}\int\sinh\,dx&=\int\frac{e^x-e^{-x}}2\,dx\\&=\frac12(e^x+e^{-x})+k&\\&=\cosh+\,k,\,k\in\mathbb R\end{aligned}$ The same for $\displaystyle \cosh$
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