Mr. Integral
Math notes/Calculus II/Integration Methods

Hyperbolic Functions and Inverse Integrals — Cheat sheet

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Core rule

cosh⁡2x−sinh⁡2x=1,(sinh⁡x)′=cosh⁡x,(cosh⁡x)′=sinh⁡x.\cosh^2x-\sinh^2x=1,\quad(\sinh x)'=\cosh x,\quad(\cosh x)'=\sinh x. ∫dxa2+x2=arsinh⁡(x/a)+C(a>0).\int\frac{dx}{\sqrt{a^2+x^2}}=\operatorname{arsinh}(x/a)+C\quad(a>0).

Watch for

Inverse hyperbolic functions are not reciprocals. Preserve the chosen real domain and inner scaling factors.