Hyperbolic functions

• Hyperbolic sine is known as sinh and is equal to (e^x - e^-x)/2.
• Hyperbolic cosine is known as cosh and is equal to (e^x + e^-x)/2.
• Hyperbolic tan is known as tanh and is equal to (e^2x - 1)/(e^2x + 1).
• When proving the results for inverse hyperbolics in term of natural logarithms (of which are given in the FB), let y = inverse hyperbolic (x) and therefore x = hyperbolic (y), then use the exponential definition and multiply by a variation of e^y to form a quadratic and solve.
• Osborn's rule says that given a trigonometric identity, the hyperbolic is the same except where there is a product (or implied product) of two sin terms, and then that term needs to be negated.
• For an integral involving root(x^2 + a^2), try the substitution x = a sinh u.
• For an integral involving root(x^2 - a^2), try the substitution x = a cosh u.
• When proving the differential for an inverse hyperbolic function, try and use the rule cosh^2 - sinh^2 = 1 to replace the differentiated result with something in terms of x.