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Undid revision 1242064119 by 67.249.248.128 (talk) "intuitive" is pretty subjective, and while it's easy to verify that it's a solution, uniqueness is less obvious |
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=== Hyperbolic cosine ===
It can be shown that the [[area under the curve]] of the hyperbolic cosine (over a finite interval) is always equal to the arc length (see ''[[
<math display="block">\text{area} = \int_a^b \cosh x \,dx = \int_a^b \sqrt{1 + \left(\frac{d}{dx} \cosh x \right)^2} \,dx = \text{arc length.}</math>
since,
<math display="block">\begin{alignat}{0}
& \cosh x = \sqrt{1 + \left(\frac{d}{dx} \cosh x \right)^2}\\
& \cosh^2x = 1 + \sinh^2x\\
& \left(\frac {e^x + e^{-x}} {2}\right)^2 = 1 + \left(\frac {e^x - e^{-x}} {2}\right)^2\\
& \frac {e^{2x} + e^{-2x} + 2} {4} = \frac {e^{2x} + e^{-2x} + 2} {4}\\
\end{alignat}</math>
===Hyperbolic tangent{{anchor|tanh}}===
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