sinhx2

Sinhx2 is not a standard named function in hyperbolic trigonometry. In informal plain-text notation, it is often used to indicate the square of the hyperbolic sine, written as sinh^2 x in mathematical notation. This article uses that interpretation unless stated otherwise.

Definition and basic form: The hyperbolic sine is defined as sinh x = (e^x - e^{-x})/2. Therefore sinh^2 x = ((e^x - e^{-x})/2)^2 = (e^{2x} - 2 + e^{-2x})/4 = (cosh 2x - 1)/2, where cosh 2x is the hyperbolic cosine of 2x.

Key identities: Using the standard hyperbolic identities, cosh 2x = cosh^2 x + sinh^2 x and the relation sinh^2 x = cosh^2 x - 1, one can derive that sinh^2 x = (cosh 2x - 1)/2. The expression can also be written purely in terms of exponentials as (e^{2x} - 2 + e^{-2x})/4.

Differentiation and integration: The derivative of sinh^2 x with respect to x is 2 sinh x cosh x, which equals sinh 2x. An antiderivative is ∫ sinh^2 x dx = (sinh 2x)/4 - x/2 + C, derived from the identity sinh^2 x = (cosh 2x - 1)/2.

Alternative interpretation note: If the notation was intended to denote sinh(x^2) (the hyperbolic sine of x squared), that function is defined as (e^{x^2} - e^{-x^2})/2 and satisfies d/dx [sinh(x^2)] = 2x cosh(x^2). It obeys different properties and does not relate to the double-angle identities for sinh^2 x.

Applications: Sinh^2 x appears in various hyperbolic-trig contexts, including integrals, area computations for hyperbolic sectors, and in expressions derived from exponential representations of hyperbolic functions.