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

[swampwiz]

I am looking at the formulae for the integral of csch( x ) at Wikipedia.

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 ) = csch²( x ) / [ csch²( x ) ]^1/2

= csch²( x ) / [ coth²( x ) - 1 ]^1/2

u = coth( x )

du = - csch²( x )

csch( x ) dx = - ( u² - 1 )^{( - 1/2 )}

∫ - ( u² - 1 )^{( - 1/2 )} = - ln | u + ( u² - 1 )^1/2 | + C

= - ln | coth( x ) + [ coth²( 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 +.

 

[slider142]

The square root of a positive real number is defined to be the positive number whose square is that positive real number only, as a function can only have one value. So the square root of csc²(x) would be the absolute value of csc(x), or |csc(x)|. As such, your first equation is false: the hyperbolic cosecant of ln(1/2) is -4/3, but the fraction on the right side of your equation gives us only 4/3.
In order to verify that an expression is an integral of another expression, we only need to differentiate it, as there are many ways to express the same quantity, and our derivation may not match the author's derivation, even though the resulting quantities may be the same up to a constant of integration.
If we differentiate ln | [ ( cosh( x ) - 1 ] / sinh( x ) ], we get csch(x), so this is a valid integral, up to a constant of integration. Whether we can find a series of integration techniques that leads us to that particular expression is a secondary issue. Sometimes, for example, but probably not in this case, the integral is merely an educated guess, whose derivative checks out, and thus must be equal to the integral up to a constant of integration, by the fundamental theorem of calculus.