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HyperbolicFunctions

HyperbolicFunctions - Hyperbolic Functions Hyperbolic...

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Unformatted text preview: Hyperbolic Functions Hyperbolic cosine of x: Hyperbolic sine of x: cosh x = sinh x = e +e 2 x βˆ’x Integrals e x βˆ’ eβˆ’ x 2 ∫ sinh u du = cosh u + C ∫ cosh u du = sinh u + C ∫ sech u du = tanh u + C 2 ∫ csch u du = βˆ’ coth u + C ∫ sech u tanh u du = βˆ’ sech u + C ∫ csch u coth u du = βˆ’ csch u + C 2 Hyperbolic tangent: Hyperbolic cotangent: Hyperbolic secant: Hyperbolic cosecant: tanh x = coth x = sech x = csch x = sinh x e x βˆ’ e βˆ’ x = cosh x e x + e βˆ’ x cosh x e x + e βˆ’ x = sinh x e x βˆ’ e βˆ’ x 1 2 =x cosh x e + e βˆ’ x 1 2 = sinh x e x βˆ’ eβˆ’ x Useful Identities sech βˆ’1 x = cosh βˆ’1 1 x csch βˆ’1 x = sinh βˆ’1 1 x coth βˆ’1 x = tanh βˆ’1 1 x Derivatives of Inverse Hyperbolic Functions d ( sinh βˆ’1 u ) dx d ( cosh u ) βˆ’1 Logarithm Formulas for Evaluating Inverse Hyperbolic Functions = = du 1 + u dx 2 1 sinh βˆ’1 x = ln x + x 2 + 1 , βˆ’ ∞ < x < ∞ cosh βˆ’1 2 Identities sinh x + cosh x = e x cosh 2 x βˆ’ sinh 2 x = 1 tanh x = 1 βˆ’ sech x 2 2 dx sinh 2 x = 2sinh x cosh x cosh 2 x = cosh 2 x + sinh 2 x cosh 2 x + 1 cosh x = 2 cosh 2 x βˆ’ 1 sinh 2 x = 2 2 d ( tanh βˆ’1 u ) dx d ( coth βˆ’1 u ) dx d ( sech βˆ’1 u ) dx d ( csch βˆ’1 u ) dx du dx du dx du , u >1 2 u βˆ’ 1 dx 1 ( x = ln ( x + ) x βˆ’ 1) , x β‰₯ 1 1 du = , u <1 1 βˆ’ u 2 dx = = = 1 du , u >1 1 βˆ’ u 2 dx du , 0 < u <1 u 1 βˆ’ u dx 2 coth x = 1 + csch x 2 2 βˆ’1 Derivatives d du ( sinh u ) = cosh u dx dx d du ( cosh u ) = sinh u dx dx d du 2 ( tanh u ) = sech u dx dx d du ( coth u ) = βˆ’csch 2u dx dx d ( sech u ) = βˆ’ sech u tanh u dx d ( cschu ) = βˆ’csch u coth u dx du ,uβ‰ 0 u 1 + u dx 2 βˆ’1 1 1+ x tanh βˆ’1 x = ln , x <1 2 1βˆ’ x 1 + 1 βˆ’ x2 sech βˆ’1 x = ln , 0 < x ≀1 x 1 1 + x2 csch βˆ’1 x = ln + , xβ‰ 0 x x 1 x +1 βˆ’1 coth x = ln , x >1 2 x βˆ’1 Integrals of Inverse Hyperbolic Functions ∫ ∫ du 1+ u du 2 = sinh βˆ’1 u + C = cosh βˆ’1 u + C , u > 1 u2 βˆ’1 βˆ’1 du tanh u + C if u < 1 = ∫ 1 βˆ’ u 2 coth βˆ’1 u + C if u > 1 1 = βˆ’sech βˆ’1 u + C = βˆ’ cosh βˆ’1 + C u 1βˆ’ u 1 du βˆ’1 βˆ’1 ∫ u 1 + u 2 = βˆ’csch u + C = βˆ’ sinh u + C ∫u du 2 ...
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