| Integrals of Hyperbolic Functions | |
| $ \int \sinh u \, du $ | $ = \cosh u + C $ |
| $ \int \cosh u \, du $ | $ = \sinh u + C $ |
| $ \int \operatorname{sech}^2 u \, du $ | $ = \tanh u + C $ |
| $ \int \operatorname{csch}^2 u \, du $ | $ = -\coth u + C $ |
| $ \int \operatorname{sech} u \tanh u \, du $ | $ = -\operatorname{sechu} + C $ |
| $ \int \operatorname{csch} u \coth u \, du $ | $ = -\operatorname{cschu} + C $ |
| Integrals Involving Inverse Hyperbolic Functions | |
| $ \int \frac{1}{\sqrt{u^2 + a^2}} \, du $ | $ = \sinh^{-1} \left( \frac{u}{a} \right) + C \quad (a > 0) $ |
| $ \int \frac{1}{\sqrt{u^2 - a^2}} \, du $ | $ = \cosh^{-1} \left( \frac{u}{a} \right) + C \quad (u > a > 0) $ |
| $ \int \frac{1}{a^2 - u^2} \, du $ | $ = \begin{cases} \frac{1}{a} \tanh^{-1} \left( \frac{u}{a} \right) + C & (u^2 < a^2) \\ \frac{1}{a} \coth^{-1} \left( \frac{u}{a} \right) + C & (u^2 > a^2) \end{cases} $ |
| $ \int \frac{1}{u\sqrt{a^2 - u^2}} \, du $ | $ = -\frac{1}{a} \operatorname{sech}^{-1} \left( \frac{u}{a} \right) + C \quad (0 < u < a) $ |
| $ \int \frac{1}{u\sqrt{a^2 + u^2}} \, du $ | $ = -\frac{1}{a} \operatorname{csch}^{-1} \left| \frac{u}{a} \right| + C $ |
| Derivatives of Inverse Hyperbolic Functions | |
| $ \frac{d}{dx} \sinh^{-1} u $ | $ = \frac{1}{\sqrt{1 + u^2}} \frac{du}{dx} $ |
| $ \frac{d}{dx} \cosh^{-1} u $ | $ = \frac{1}{\sqrt{u^2 - 1}} \frac{du}{dx} \quad (u > 1) $ |
| $ \frac{d}{dx} \tanh^{-1} u $ | $ = \frac{1}{1 - u^2} \frac{du}{dx} \quad (|u| < 1) $ |
| $ \frac{d}{dx} \operatorname{csch}^{-1} u $ | $ = \frac{-1}{|u|\sqrt{1 + u^2}} \frac{du}{dx} \quad (u \neq 0) $ |
| $ \frac{d}{dx} \operatorname{sech}^{-1} u $ | $ = \frac{-1}{u\sqrt{1 - u^2}} \frac{du}{dx} \quad (0 < u < 1) $ |
| $ \frac{d}{dx} \coth^{-1} u $ | $ = \frac{1}{1 - u^2} \frac{du}{dx} \quad (|u| > 1) $ |
| Expressing Inverse Hyperbolic Functions As Natural Logarithms | |
| $ \sinh^{-1} x $ | $ = \ln \left( x + \sqrt{x^2 + 1} \right) \quad (-\infty < x < \infty) $ |
| $ \cosh^{-1} x $ | $ = \ln \left( x + \sqrt{x^2 - 1} \right) \quad (x \geq 1) $ |
| $ \tanh^{-1} x $ | $ = \frac{1}{2} \ln \left( \frac{1 + x}{1 - x} \right) \quad (|x| < 1) $ |
| $ \operatorname{sech}^{-1} x $ | $ = \ln \left( \frac{1 + \sqrt{1 - x^2}}{x} \right) \quad (0 < x \leq 1) $ |
| $ \operatorname{csch}^{-1} x $ | $ = \ln \left( \frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|} \right) \quad (x \neq 0) $ |
| $ \coth^{-1} x $ | $ = \frac{1}{2} \ln \left( \frac{x + 1}{x - 1} \right) \quad (|x| > 1) $ |
| Alternate Form For Integrals Involving Inverse Hyperbolic Functions | |
| $ \int \frac{1}{\sqrt{u^2 \pm a^2}} \, du $ | $ = \ln \left( u + \sqrt{u^2 \pm a^2} \right) + C $ |
| $ \int \frac{1}{a^2 - u^2} \, du $ | $ = \frac{1}{2a} \ln \left| \frac{a + u}{a - u} \right| + C $ |
| $ \int \frac{1}{u\sqrt{a^2 \pm u^2}} \, du $ | $ = -\frac{1}{a} \ln \left( \frac{a + \sqrt{a^2 \pm u^2}}{|u|} \right) + C $ |