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 −x Integrals e x − e− x 2 ∫ sinh u du = cosh u + C ∫ cosh u du = sinh u + C ∫ sech u du = tanh u + C 2 ∫ csch u du = − coth u + C ∫ sech u tanh u du = − sech u + C ∫ csch u coth u du = − csch u + C 2 Hyperbolic tangent: Hyperbolic cotangent: Hyperbolic secant: Hyperbolic cosecant: tanh x = coth x = sech x = csch x = sinh x e x − e − x = cosh x e x + e − x cosh x e x + e − x = sinh x e x − e − x 1 2 =x cosh x e + e − x 1 2 = sinh x e x − e− x Useful Identities sech −1 x = cosh −1 1 x csch −1 x = sinh −1 1 x coth −1 x = tanh −1 1 x Derivatives of Inverse Hyperbolic Functions d ( sinh −1 u ) dx d ( cosh u ) −1 Logarithm Formulas for Evaluating Inverse Hyperbolic Functions = = du 1 + u dx 2 1 sinh −1 x = ln x + x 2 + 1 , − ∞ < x < ∞ cosh −1 2 Identities sinh x + cosh x = e x cosh 2 x − sinh 2 x = 1 tanh x = 1 − 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 − 1 sinh 2 x = 2 2 d ( tanh −1 u ) dx d ( coth −1 u ) dx d ( sech −1 u ) dx d ( csch −1 u ) dx du dx du dx du , u >1 2 u − 1 dx 1 ( x = ln ( x + ) x − 1) , x ≥ 1 1 du = , u <1 1 − u 2 dx = = = 1 du , u >1 1 − u 2 dx du , 0 < u <1 u 1 − u dx 2 coth x = 1 + csch x 2 2 −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 ) = −csch 2u dx dx d ( sech u ) = − sech u tanh u dx d ( cschu ) = −csch u coth u dx du ,u≠0 u 1 + u dx 2 −1 1 1+ x tanh −1 x = ln , x <1 2 1− x 1 + 1 − x2 sech −1 x = ln , 0 < x ≤1 x 1 1 + x2 csch −1 x = ln + , x≠0 x x 1 x +1 −1 coth x = ln , x >1 2 x −1 Integrals of Inverse Hyperbolic Functions ∫ ∫ du 1+ u du 2 = sinh −1 u + C = cosh −1 u + C , u > 1 u2 −1 −1 du tanh u + C if u < 1 = ∫ 1 − u 2 coth −1 u + C if u > 1 1 = −sech −1 u + C = − cosh −1 + C u 1− u 1 du −1 −1 ∫ u 1 + u 2 = −csch u + C = − sinh u + C ∫u du 2 ...
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This note was uploaded on 02/12/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.

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