HyperbolicFunctions

HyperbolicFunctions - Hyperbolic Functions Hyperbolic...

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Unformatted text preview: Hyperbolic Functions Hyperbolic cosine of x: Hyperbolic sine of x: cosh x = e +e 2 x -x Integrals sinh u du = cosh u + C cosh u du = sinh u + C sech 2 u du = tanh u + C csch 2 u du = - coth u + C sech u tanh u du = - sech u + C csch u coth u du = - csch u + C e x - e- x sinh x = 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 -1x = cosh -1 1 x csch -1x = sinh -1 1 x coth -1x = tanh -1 1 x Derivatives of Inverse Hyperbolic Functions d ( sinh -1 u ) dx d ( cosh -1 u ) dx = = du 1 + u dx 2 Logarithm Formulas for Evaluating Inverse Hyperbolic Functions 1 sinh -1 x = ln x + x 2 + 1 , - cosh -1 2 Identities sinh x + cosh x = e x cosh 2 x - sinh 2 x = 1 tanh x = 1 - sech x 2 2 sinh 2x = 2 sinh x cosh x cosh 2x = cosh 2 x + sinh 2 x cosh 2x + 1 cosh x = 2 cosh 2x - 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 < 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 u 1 + u dx 2 -1 0 1 1+ x tanh -1 x = ln , x <1 2 1- x + 1- x 2 1 sech -1 x = ln , 0< x 1 x 1 1+ x 2 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 tanh -1 u + C if u < 1 du = coth -1 u + C if u > 1 1- u 2 1 = -sech -1 u + C = - cosh -1 C + u 1- u du 1 -1 -1 + 2u 1+ u 2 = -csch u + C = - sinh u C 2u du 2 ...
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This note was uploaded on 03/10/2008 for the course MATH 215 taught by Professor Riggs during the Fall '05 term at Cal Poly Pomona.

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