I am looking at the formulae for the integral of csch( x ) at Wikipedia.(adsbygoogle = window.adsbygoogle || []).push({});

http://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions

and it seems that there is a mistake, specially the solution of (presume a is 1 here for simplicity)

ln | [ ( cosh( x ) - 1 ] / sinh( x ) ]

while there is also

ln | sinh( x ) / [ ( cosh( x ) + 1 ] |

the latter of which I get. The issue is that the solution to this latter one results in the former, only with terms swapped around the division operator.

Here is my derivation for them:

csch( x ) = csch^{2}( x ) / [ csch^{2}( x ) ]^{1/2}

= csch^{2}( x ) / [ coth^{2}( x ) - 1 ]^{1/2}

u = coth( x )

du = - csch^{2}( x )

csch( x ) dx = - ( u^{2}- 1 )^{( - 1/2 )}

∫ - ( u^{2}- 1 )^{( - 1/2 )}= - ln | u + ( u^{2}- 1 )^{1/2}| + C

= - ln | coth( x ) + [ coth^{2}( x ) - 1 ]^{1/2}| + C

= - ln | coth( x ) + [ { csch2( x ) } ]1/2 | + C = - ln | coth( x ) + csch( x ) | + C

= - ln | [ cosh( x ) / sinh( x ) ] + [ 1 / sinh( x ) ] | + C

= - ln | [ cosh( x ) + 1 ] / sinh( x ) | + C

= ln | sinh( x ) / [ cosh( x ) + 1 ] | + C

So I get one the solutions, but not the other, which as I had said, seems to have the terms swapped around the division operator.

I presume that it is proper to consider the square root of something to be +.

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# I think there is a mistake on Wikipedia about the integral of csch

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