Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverses of hyperbolic functions
They can be expressed in logarithmic form:
- $\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})$
- $\cosh^{-1} x = \ln(x + \sqrt{x^2 - 1})$, $x \ge 1$
- $\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$, $|x| < 1$
- $\text{sech}^{-1} x = \cosh^{-1}(1/x)$

Derivatives follow formulas similar to inverse trig functions:
- $\frac{d}{dx} [\sinh^{-1} u] = \frac{1}{\sqrt{u^2 + 1}} \cdot \frac{du}{dx}$
- $\frac{d}{dx} [\cosh^{-1} u] = \frac{1}{\sqrt{u^2 - 1}} \cdot \frac{du}{dx}$, $u > 1$
- $\frac{d}{dx} [\tanh^{-1} u] = \frac{1}{1 - u^2} \cdot \frac{du}{dx}$, $|u| < 1$
- $\frac{d}{dx} [\text{sech}^{-1} u] = -\frac{1}{u\sqrt{1-u^2}} \cdot \frac{du}{dx}$
Example: $y = \cosh^{-1}(5x - 7)$ → $\frac{dy}{dx} = \frac{5}{\sqrt{(5x-7)^2 - 1}}$

Key standard integrals derived from inverse hyperbolic derivatives:
- $\int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1}\left(\frac{x}{a}\right) + C = \ln(x + \sqrt{x^2 + a^2}) + C$
- $\int \frac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}\left(\frac{x}{a}\right) + C = \ln(x + \sqrt{x^2 - a^2}) + C$
- $\int \frac{dx}{a^2 - x^2} = \frac{1}{a} \tanh^{-1}\left(\frac{x}{a}\right) + C = \frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C$, $|x| < a$
Applications: Substitute into forms like $\sqrt{1+9x^2}$, $\sqrt{9x^2 - 25}$, $x\sqrt{1+4x^2}$