Trigonometric Function

Hyperbolic Function

sinh⁡z=ez−e−z2=−isin⁡iz\sinh z = \frac{e^z - e^{-z}}{2} = -i \sin iz

cosh⁡z=ez+e−z2=cos⁡iz\cosh z = \frac{e^z + e^{-z}}{2} = \cos iz

tanh⁡z=ez−e−zez+e−z=−itan⁡iz\tanh z = \frac{e^z - e^{-z}}{e^z + e^{-z}} = -i \tan iz

coth⁡z=ez+e−zez−e−z=icot⁡iz\coth z = \frac{e^z + e^{-z}}{e^z - e^{-z}} = i \cot iz

sech⁡z=2ez+e−z=sec⁡iz\operatorname{sech} z = \frac{2}{e^z + e^{-z}} = \sec iz

csch⁡z=2ez−e−z=icsc⁡iz\operatorname{csch} z = \frac{2}{e^z - e^{-z}} = i \csc iz

Inverse Hyperbolic Function

arsinh⁡z=Ln⁡(z+z2+1)\operatorname{arsinh} z = \operatorname{Ln}(z + \sqrt{z^2 + 1})

arcosh⁡z=Ln⁡(z+z+1z−1)\operatorname{arcosh} z = \operatorname{Ln}(z + \sqrt{z + 1} \sqrt{z - 1})

artanh⁡z=12Ln⁡(1+z1−z)\operatorname{artanh} z = \frac{1}{2} \operatorname{Ln}(\frac{1 + z}{1 - z})

arcoth⁡z=12Ln⁡(z+1z−1)\operatorname{arcoth} z = \frac{1}{2} \operatorname{Ln}(\frac{z + 1}{z - 1})

arsech⁡z=Ln⁡(1z+1z+11z−1)\operatorname{arsech} z = \operatorname{Ln}(\frac{1}{z} + \sqrt{\frac{1}{z} + 1} \sqrt{\frac{1}{z} - 1})

arcsch⁡z=Ln⁡(1z+1z2+1)\operatorname{arcsch} z = \operatorname{Ln}(\frac{1}{z} + \sqrt{\frac{1}{z^2} + 1})

Differentiation

(sin⁡x)′=cos⁡x(\sin x)' = \cos x

(cos⁡x)′=−sin⁡x(\cos x)' = -\sin x

(tan⁡x)′=sec⁡2x(\tan x)' = \sec^2 x

(cot⁡x)′=−csc⁡2x(\cot x)' = -\csc^2 x

(sec⁡x)′=sec⁡xtan⁡x(\sec x)' = \sec x \tan x

(csc⁡x)′=−csc⁡xcot⁡x(\csc x)' = -\csc x \cot x

(sinh⁡x)′=cosh⁡x(\sinh x)' = \cosh x

(cosh⁡x)′=sinh⁡x(\cosh x)' = \sinh x

(tanh⁡x)′=sech⁡2x(\tanh x)' = \operatorname{sech}^2 x

(coth⁡x)′=−csch⁡2x(\coth x)' = -\operatorname{csch}^2 x

(sech⁡x)′=−sech⁡xtanh⁡x(\operatorname{sech} x)' = -\operatorname{sech} x \tanh x

(csch⁡x)′=−csch⁡xcoth⁡x(\operatorname{csch} x)' = -\operatorname{csch} x \coth x

(arcsin⁡x)′=11−x2(\arcsin x)' = \frac{1}{\sqrt{1 - x^2}}

(arccos⁡x)′=−11−x2(\arccos x)' = -\frac{1}{\sqrt{1 - x^2}}

(arctan⁡x)′=11+x2(\arctan x)' = \frac{1}{1 + x^2}

(arccot⁡x)′=−11+x2(\operatorname{arccot} x)' = -\frac{1}{1 + x^2}

(arcsec⁡x)′=1∣x∣x2−1(\operatorname{arcsec} x)' = \frac{1}{|x| \sqrt{x^2 - 1}}

(arccsc⁡x)′=−1∣x∣x2−1(\operatorname{arccsc} x)' = -\frac{1}{|x| \sqrt{x^2 - 1}}

(arsinh⁡x)′=1x2+1(\operatorname{arsinh} x)' = \frac{1}{\sqrt{x^2 + 1}}

(arcosh⁡x)′=1x2−1(\operatorname{arcosh} x)' = \frac{1}{\sqrt{x^2 - 1}}

(artanh⁡x)′=11−x2(\operatorname{artanh} x)' = \frac{1}{1 - x^2}

(arcoth⁡x)′=11−x2(\operatorname{arcoth} x)' = \frac{1}{1 - x^2}

(arsech⁡x)′=−1x1−x2(\operatorname{arsech} x)' = -\frac{1}{x \sqrt{1 - x^2}}

(arcsch⁡x)′=−1∣x∣1+x2(\operatorname{arcsch} x)' = -\frac{1}{|x| \sqrt{1 + x^2}}

Integration

∫sin⁡x dx=−cos⁡x+C\int \sin x \ dx = -\cos x + C

∫cos⁡x dx=sin⁡x+C\int \cos x \ dx = \sin x + C

∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int \tan x \ dx = -\ln |\cos x| + C

∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x \ dx = \ln |\sin x| + C

∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C\int \sec x \ dx = \ln |\sec x + \tan x| + C

∫csc⁡x dx=−ln⁡∣csc⁡x+cot⁡x∣+C\int \csc x \ dx = -\ln |\csc x + \cot x| + C

∫sinh⁡x dx=cosh⁡x+C\int \sinh x \ dx = \cosh x + C

∫cosh⁡x dx=sinh⁡x+C\int \cosh x \ dx = \sinh x + C

∫tanh⁡x dx=ln⁡(cosh⁡x)+C\int \tanh x \ dx = \ln(\cosh x) + C

∫coth⁡x dx=ln⁡∣sinh⁡x∣+C\int \coth x \ dx = \ln|\sinh x| + C

∫sech⁡x dx=arctan⁡(sinh⁡x)+C\int \operatorname{sech} x \ dx = \arctan(\sinh x) + C

∫csch⁡x dx=−ln⁡∣csch⁡x+coth⁡x∣+C\int \operatorname{csch} x \ dx = -\ln|\operatorname{csch} x + \coth x| + C

∫arcsin⁡x dx=xarcsin⁡x+1−x2+C\int \arcsin x \ dx = x \arcsin x + \sqrt{1 - x^2} + C

∫arccos⁡x dx=xarccos⁡x−1−x2+C\int \arccos x \ dx = x \arccos x - \sqrt{1 - x^2} + C

∫arctan⁡x dx=xarctan⁡x−12ln⁡(1+x2)+C\int \arctan x \ dx = x \arctan x - \frac{1}{2} \ln(1 + x^2) + C

∫arccot⁡x dx=xarccot⁡x+12ln⁡(1+x2)+C\int \operatorname{arccot} x \ dx = x \operatorname{arccot} x + \frac{1}{2} \ln(1 + x^2) + C

∫arcsec⁡x dx=xarcsec⁡x−arcosh⁡∣x∣+C\int \operatorname{arcsec} x \ dx = x \operatorname{arcsec} x - \operatorname{arcosh} |x| + C

∫arccsc⁡x dx=xarccsc⁡x+arcosh⁡∣x∣+C\int \operatorname{arccsc} x \ dx = x \operatorname{arccsc} x + \operatorname{arcosh} |x| + C

∫arsinh⁡x dx=xarsinh⁡x−x2+1+C\int \operatorname{arsinh} x \ dx = x \operatorname{arsinh} x - \sqrt{x^2 + 1} + C

∫arcosh⁡x dx=xarcosh⁡x−x2−1+C\int \operatorname{arcosh} x \ dx = x \operatorname{arcosh} x - \sqrt{x^2 - 1} + C

∫artanh⁡x dx=xartanh⁡x+12ln⁡(1−x2)+C\int \operatorname{artanh} x \ dx = x \operatorname{artanh} x + \frac{1}{2} \ln(1 - x^2) + C

∫arcoth⁡x dx=xarcoth⁡x+12ln⁡(x2−1)+C\int \operatorname{arcoth} x \ dx = x \operatorname{arcoth} x + \frac{1}{2} \ln(x^2 - 1) + C

∫arsech⁡x dx=xarsech⁡x+arcsin⁡x+C\int \operatorname{arsech} x \ dx = x \operatorname{arsech} x + \arcsin x + C

∫arcsch⁡x dx=xarcsch⁡x+arsinh⁡∣x∣+C\int \operatorname{arcsch} x \ dx = x \operatorname{arcsch} x + \operatorname{arsinh} |x| + C