2.3.1—Hyperbolic functions

The core derivatives are d(sinh x)/dx=cosh x, d(cosh x)/dx=sinh x and d(tanh x)/dx=sech²x. Their integrals follow by reversing these relationships, with a constant of integration.

Use the chain rule for sinh(ax+b), and remember that ∫sech²x dx=tanh x. For logarithmic forms, check the domain and absolute-value convention where relevant.

∫3sinh(3x+1)dx=cosh(3x+1)+C, because the inner derivative 3 cancels the coefficient.

The derivative of cosh x is +sinh x, not −sinh x, and tanh is not differentiated like ordinary tan x.