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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 βx Integrals e x β eβ x 2 β« sinh u du = cosh u + C β« cosh u du = sinh u + C β« sech u du = tanh u + C 2 β« csch u du = β coth u + C β« sech u tanh u du = β sech u + C β« csch u coth u du = β csch u + C 2 Hyperbolic tangent: Hyperbolic cotangent: Hyperbolic secant: Hyperbolic cosecant: tanh x = coth x = sech x = csch x = sinh x e x β e β x = cosh x e x + e β x cosh x e x + e β x = sinh x e x β e β x 1 2 =x cosh x e + e β x 1 2 = sinh x e x β eβ x Useful Identities sech β1 x = cosh β1 1 x csch β1 x = sinh β1 1 x coth β1 x = tanh β1 1 x Derivatives of Inverse Hyperbolic Functions d ( sinh β1 u ) dx d ( cosh u ) β1 Logarithm Formulas for Evaluating Inverse Hyperbolic Functions = = du 1 + u dx 2 1 sinh β1 x = ln x + x 2 + 1 , β β < x < β cosh β1 2 Identities sinh x + cosh x = e x cosh 2 x β sinh 2 x = 1 tanh x = 1 β 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 β 1 sinh 2 x = 2 2 d ( tanh β1 u ) dx d ( coth β1 u ) dx d ( sech β1 u ) dx d ( csch β1 u ) dx du dx du dx du , u >1 2 u β 1 dx 1 ( x = ln ( x + ) x β 1) , x β₯ 1 1 du = , u <1 1 β u 2 dx = = = 1 du , u >1 1 β u 2 dx du , 0 < u <1 u 1 β u dx 2 coth x = 1 + csch x 2 2 β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 ) = βcsch 2u dx dx d ( sech u ) = β sech u tanh u dx d ( cschu ) = βcsch u coth u dx du ,uβ 0 u 1 + u dx 2 β1 1 1+ x tanh β1 x = ln , x <1 2 1β x 1 + 1 β x2 sech β1 x = ln , 0 < x β€1 x 1 1 + x2 csch β1 x = ln + , xβ 0 x x 1 x +1 β1 coth x = ln , x >1 2 x β1 Integrals of Inverse Hyperbolic Functions β« β« du 1+ u du 2 = sinh β1 u + C = cosh β1 u + C , u > 1 u2 β1 β1 du tanh u + C if u < 1 = β« 1 β u 2 coth β1 u + C if u > 1 1 = βsech β1 u + C = β cosh β1 + C u 1β u 1 du β1 β1 β« u 1 + u 2 = βcsch u + C = β sinh u + C β«u du 2 ...
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