The integral is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int_0^{f(E)} \frac{\cosh \alpha x}{\sqrt{1-\beta \cosh^2 \alpha x}} dx,

[/tex]

where

[tex]

f(E) = \frac{1}{\alpha}\cosh^{-1}(\sqrt{\frac{1}{\beta}}),

[/tex]

[tex]\alpha > 0[/tex], and [tex]0<\beta<1[/tex]. The integral has proven very difficult to evaluate. Every time I plug it into Mathematica with values for [tex]\alpha[/tex] and [tex]\beta[/tex] that satisfy the conditions listed above, I get an imaginary number! But that can't happen because the integral is involved in computing the period of a particle undergoing periodic motion, which is necessarily real.

It is possible that the integral will not evaluate to a real number, in which case I've made a mistake in deriving its form. If that's the case, please let me know. Thanks!

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# Could someone explain how to evaluate this horrible integral?

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