Question

Hyperbolic Trigonometric Substitution Example :)

Answer 1

Suppose you wanted to integrate a function in the form:
\[\int\limits \frac{dx}{\sqrt{x^2+\psi^2}}\]

Answer 2

Make the substitution:\[x=\psi \sinh(x) \implies dx=\psi \cosh(x)\]
So we arrive at:
\[\int\limits \frac{\psi \cosh(\zeta) d \zeta}{\sqrt{\psi^2(\sinh^2(\zeta)+1)}}=\int\limits \frac{\cosh(\zeta)d \zeta}{\sqrt{\cosh^2(\zeta)}}=\int\limits d \zeta=\zeta=\sinh^{-1}(\frac{x}{\psi})\]

Answer 3

Or:
\[\sinh^{-1}(\frac{x}{\psi})=\ln \left[ \frac{1}{\psi} \left( x+\sqrt{\left( \psi^2+x^2\right)}\right) \right]\]