Inverse hyperbolic functions

Inverse Hyperbolic Functions and Their Properties

• The inverse of the hyperbolic sine function is denoted as arsinh(x) or sinh^-1(x). This function gives the number whose hyperbolic sine is a given number.
• The inverse of the hyperbolic cosine function is denoted as arcosh(x) or cosh^-1(x). This function gives the number whose hyperbolic cosine is a specified number.
• The inverse of the hyperbolic tangent function is denoted as artanh(x) or tanh^-1(x). This function provides the number whose hyperbolic tangent is a certain number.
• Inverse hyperbolic functions also exist for sech(x), csch(x), and coth(x), and are represented respectively as sech^-1(x), csch^-1(x), and coth^-1(x).
• Just like hyperbolic functions, inverse hyperbolic functions also have specific domain and range specifications. For example, the domain of cosh^-1(x) is [1, ∞) and the range is [0, ∞).

Derivative Forms for Inverse Hyperbolic Functions

• The derivative of arsinh(x) with respect to x, represented as d/dx(arsinh x), is equal to 1/√(1 + x^2).
• The derivative of arcosh(x) with respect to x, denoted as d/dx(arcosh x), is 1/√(x^2 - 1), for x > 1.
• The derivative of artanh(x) with respect to x, denoted d/dx(artanh x), equals 1/(1 - x^2), for -1 < x < 1.
• The derivative forms for sech^-1(x), csch^-1(x), and coth^-1(x) can be deduced similarly.

Integral Forms involving Inverse Hyperbolic Functions

• Be familiar with the integral forms involving inverse hyperbolic functions. For instance, the integral of arsinh x dx, denoted as ∫arsinh x dx, becomes x*arsinh x - √(1 + x^2) + C.
• The ability to integrate functions with inverse hyperbolic functions via substitution or other integration methods is essential. Therefore, tackling integrals like ∫x arsinh x dx or ∫x arcosh x dx requires a command of integration skills similar to those applied in integrals with trigonometric functions.
• Recall, the arbitrary constant ‘C,’ known as the constant of integration, is added to all indefinite integrals considering the derivative of any constant is zero.