Integration of Hyperbolic & Inverse Hyperbolic Functions

Key Identities
- $\frac{d}{dx} \sinh x = \cosh x$
- $\frac{d}{dx} \cosh x = \sinh x$
- $\frac{d}{dx} \tanh x = \operatorname{sech}^2 x$
- $\frac{d}{dx} \sinh^{-1} x = \frac{1}{\sqrt{x^2+1}}$
- $\frac{d}{dx} \cosh^{-1} x = \frac{1}{\sqrt{x^2-1}}$
- $\frac{d}{dx} \tanh^{-1} x = \frac{1}{1-x^2}$
Integration by Substitution
- Use hyperbolic functions when integrand contains $\sqrt{x^2 \pm a^2}$
  - Substitute $x = a \sinh t$ for $\sqrt{x^2 + a^2}$
  - Substitute $x = a \cosh t$ for $\sqrt{x^2 - a^2}$
- Example: $\int \frac{e^x}{\sqrt{e^{2x}+1}} , dx$ → let $u = e^x$, then $\int \frac{du}{\sqrt{u^2+1}} = \sinh^{-1} u + C$
Inverse Hyperbolic Integrals
- $\int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1}(\frac{x}{a}) + C$
- $\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}(\frac{x}{a}) + C$ (for $x > a$)
- $\int \frac{dx}{a^2 - x^2} = \frac{1}{a} \tanh^{-1}(\frac{x}{a}) + C$ (for $|x| < a$)
Examples from Tutorial
- $\int \frac{\sinh^{-1}(\sqrt{x})}{x^{3/2}} , dx$ — use substitution $u = \sinh^{-1}(\sqrt{x})$
- $\int \frac{\sinh^{-1}(3x)}{\sqrt{9x^2+1}} , dx$ — let $u = \sinh^{-1}(3x)$, $du = \frac{3}{\sqrt{9x^2+1}} dx$
- $\int \frac{\cosh^{-1}(2x)}{\sqrt{4x^2-1}} , dx$ — let $u = \cosh^{-1}(2x)$
- $\int x \sqrt{(\ln 2x)^2 + 9} , dx$ — substitute $u = \ln 2x$