Math 2414 Section 6 10 Hyperbolic Functions Hyperbolic Functions When we construct trigonometric like functions with respect to a hyperbola rather than a circle x x the result is a new family of six functions that involve e and e They are called the hyperbolic trigonometric functions Definition Hyperbolic Functions Hyperbolic sine Hyperbolic cosine Hyperbolic tangent Hyperbolic cotangent Hyperbolic secant Hyperbolic cosecant sinh x e x e x 2 cosh x tanh x e x e x 2 sinh x e x e x cosh x e x e x cosh x e x e x coth x sinh x e x e x 1 2 sech x x x cosh x e e 1 2 csch x x x sinh x e e Hyperbolic Identities cosh 2 x sinh 2 x 1 1 tanh 2 x sech 2 x coth 2 x 1 csch 2 x sinh x sinh x cosh x cosh x tanh x tanh x sinh 2 x 2sinh x cosh x cosh 2 x cosh 2 x sinh 2 x cosh 2 x 1 2 cosh 2 x 1 cosh 2 x 2 sinh x y sinh x cosh y cosh x sinh y sinh 2 x cosh x y cosh x cosh y sinh x sinh y Derivative and Integral Formulas of Hyperbolic Functions d sinh x cosh x cosh x dx sinh x C 1 dx d cosh x sinh x sinh x dx cosh x C 2 dx d tanh x sech 2 x sech 2 x dx tanh x C dx 3 d csch x csch x coth x csch x coth x dx csch x C 4 dx d sech x sech x tanh x sech x tanh x dx sech x C 5 dx d coth x csch 2 x csch 2 x dx coth x C dx 6 d sinh x cosh x Prove dx d sech 3x Evaluate dx d2 sech 3 x 2 Evaluate dx ln 3 Evaluate sinh 0 3 x cosh x dx ln 3 Evaluate sech x dx 0
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