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Evaluate csch(x)
Evaluate sech(x)
Evaluate arsech(x)
Evaluate exponential
Evaluate arccsc(x)
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Inverse hyperbolic cosecant calculator

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

This tool evaluates the inverse hyperbolic cosecant of a number: arcsch(x). Enter the argument x below.

x =
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arcsch(x) =
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Definitions

General

The inverse hyperbolic cosecant function, in modern notation written as arcsch(x) or arccsch(x) or csch-1x, gives the value t (hyperbolic angle), so that:

\mathrm{csch}\,{t} = x

The inverse hyperbolic cosecant function accepts as arguments all real numbers except zero. Since the hyperbolic cosecant is defined through the natural exponential function \mathrm{e}^x, its inverse can be defined through the natural logarithm function, using the following formula, for real x, with x≠1:

\mathrm{arcsch}\,{x} = \ln\left(\frac{1+\sqrt{1+x^2}}{x}\right)

Properties

The derivative of the inverse hyperbolic cosecant function is:

\left(\mathrm{arcsch}\,{x}\right)' = \frac{-1}{|x|\sqrt{1+x^2}}\quad, x\ne 0

The integral of the inverse hyperbolic cosecant function is given by:

\int \mathrm{arcsch}\,{x}\, \mathrm{d}x = x\, \mathrm{arcsch}\,{x} + \mathrm{arcoth}\,\left(\sqrt{\frac{1}{x^2}+1} \right) + C \quad, x\ne 0

See also
Evaluate csch(x)
Evaluate sech(x)
Evaluate arsech(x)
Evaluate exponential
Evaluate arccsc(x)
All evaluation tools