User:Arrristotle2600/Trig Calc Tables

This is a list of the 24 trigonometric functions and their derivatives and anti-derivatives. In the anti-derivatives, the constant of integration has been omitted for brevity.

$$ \begin{array}{|c||c|} Circular Forward & Circular Inverse\\ \hline \begin{array}{c|c|c} f'(x) & f(x) & F(x)\\ \hline \cos{x}        & \sin{x} & -\cos{x}\\ -\sin{x}       & \cos{x} &  \sin{x}\\ \sec^2{x}      & \tan{x} & -\ln{\cos{x}}\\ \sec{x}\tan{x} & \sec{x} & \ln{\left|\sec{x} + \tan{x}\right|}\\ \csc{x}\cot{x} & \csc{x} & -\ln{\left|\csc{x} + \cot{x}\right|}\\ -\csc^2{x}     & \cot{x} & \ln{\left|\sin{x}\right|}\\ \end{array}&

\begin{array}{c|c|c} f'(x) & f(x) & F(x)\\ \hline \frac{1}{\sqrt{1-x^2}}      & \sin^{-1}{x} & x \, \sin^{-1}(x) + \sqrt{1 - x^2}\\ \frac{-1}{\sqrt{1-x^2}}     & \cos^{-1}{x} & x \, \cos^{-1}(x) - \sqrt{1 - x^2}\\ \frac{1}{x^2+1}             & \tan^{-1}{x} & x \, \tan^{-1}(x) - \frac{1}{2} \ln{\left| 1 + x^2\right|}\\ {1 \over |x|\sqrt{x^2 - 1}} & \sec^{-1}{x} & x \, \sec^{-1}(x) - \tanh^{-1}\,\sqrt{1-\frac{1}{x^2}}\\ -{1 \over |x|\sqrt{x^2 - 1}} & \csc^{-1}{x} & x \, \csc^{-1}(x) + \tanh^{-1}\,\sqrt{1-\frac{1}{x^2}}\\ -{1 \over 1 + x^2}          & \cot^{-1}{x} & x \, \cot^{-1}(x) + \frac{1}{2} \ln{\left| 1 + x^2\right|}\\ \end{array} \end{array} $$

$$ \begin{array}{|c||c|} Hyperbolic Forward & Hyperbolic Inverse\\ \hline \begin{array}{c|c|c} f'(x) & f(x) & F(x)\\ \hline \cosh{x}                                         & \sinh{x}               & \cosh{x}\\ \sinh{x}                                         & \cosh{x}               & \sinh{x}\\ {\operatorname{sech}^2\,x}                       & \tanh{x}               & \ln \cosh x\\ - \tanh x\,\operatorname{sech}\,x                & \operatorname{sech}\,x & \sin^{-1}\,(\tanh x)\\ -\,\operatorname{coth}\,x\,\operatorname{csch}\,x & \operatorname{csch}\,x & \ln\left| \tanh {x \over2}\right|\\ -\,\operatorname{csch}^2\,x                      & \operatorname{coth}\,x & \ln| \sinh x |\\ \end{array}&

\begin{array}{c|c|c} f'(x) & f(x) & F(x)\\ \hline { 1 \over \sqrt{x^2 + 1}}   & \operatorname{sinh}^{-1}\,x & x \, \operatorname{sinh}^{-1} \, x-\sqrt{x^2+1}\\ { 1 \over \sqrt{x^2-1}}     & \operatorname{cosh}^{-1}\,x & x \, \operatorname{cosh}^{-1} \, x-\sqrt{x+1} \, \sqrt{x-1}\\ { 1 \over 1 - x^2}          & \operatorname{tanh}^{-1}\,x & x \, \operatorname{tanh}^{-1} \, x+\frac{\ln\left(1-x^2\right)}{2}\\ -{1 \over x\sqrt{1 - x^2}}  & \operatorname{sech}^{-1}\,x & x \, \operatorname{sech}^{-1} \, x-2 \, \tan^{-1}\sqrt{\frac{1-x}{1+x}}\\ -{1 \over |x|\sqrt{1 + x^2}} & \operatorname{csch}^{-1}\,x & x \, \operatorname{csch}^{-1} \, x+\operatorname{tanh}^{-1}\sqrt{\frac{1}{x^2}+1}\\ { 1 \over 1 - x^2}          & \operatorname{coth}^{-1}\,x & x \, \operatorname{coth}^{-1} \, x+\frac{\ln\left(1-x^2\right)}{2}\\ \end{array} \end{array} $$