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Integrating Hyperbolic Functions 1

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       16/07/2018 14:11:09
There are various equivalent ways for defining the hyperbolic functions. They may be defined in terms of the exponential function:

Hyperbolic sine:
{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.} {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.}
Hyperbolic cosine:
{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.} {\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.}
Hyperbolic tangent:
{\displaystyle \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}=} \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}=
{\displaystyle ={\frac {e^{2x}-1}{e^{2x}+1}}={\frac {1-e^{-2x}}{1+e^{-2x}}}.} {\displaystyle ={\frac {e^{2x}-1}{e^{2x}+1}}={\frac {1-e^{-2x}}{1+e^{-2x}}}.}
Hyperbolic cotangent:
{\displaystyle \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}=} \coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}=
{\displaystyle ={\frac {e^{2x}+1}{e^{2x}-1}}={\frac {1+e^{-2x}}{1-e^{-2x}}},\qquad x\neq 0.} {\displaystyle ={\frac {e^{2x}+1}{e^{2x}-1}}={\frac {1+e^{-2x}}{1-e^{-2x}}},\qquad x\neq 0.}
Hyperbolic secant:
{\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}=} {\displaystyle \operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}=}
{\displaystyle ={\frac {2e^{x}}{e^{2x}+1}}={\frac {2e^{-x}}{1+e^{-2x}}}.} {\displaystyle ={\frac {2e^{x}}{e^{2x}+1}}={\frac {2e^{-x}}{1+e^{-2x}}}.}
Hyperbolic cosecant:
{\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}=} {\displaystyle \operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}=}
{\displaystyle ={\frac {2e^{x}}{e^{2x}-1}}={\frac {2e^{-x}}{1-e^{-2x}}},\qquad x\neq 0.} {\displaystyle ={\frac {2e^{x}}{e^{2x}-1}}={\frac {2e^{-x}}{1-e^{-2x}}},\qquad x\neq 0.}
The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the unique solution (s, c) of the system

{\displaystyle {\begin{aligned}c (x)&=s(x)\\s (x)&=c(x)\end{aligned}}} {\displaystyle {\begin{aligned}c (x)&=s(x)\\s (x)&=c(x)\end{aligned}}}
such that s(0) = 0 and c(0) = 1.

They are also the unique solution of the equation {\displaystyle f (x)=f(x),} {\displaystyle f (x)=f(x),} such that {\displaystyle f(0)=1,f (0)=0,} {\displaystyle f(0)=1,f (0)=0,} for the hyperbolic cosine, and {\displaystyle f(0)=0,f (0)=1,} {\displaystyle f(0)=0,f (0)=1,} for the hyperbolic sine.

Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:

Hyperbolic sine:
{\displaystyle \sinh x=-i\sin(ix)} \sinh x=-i\sin(ix)
Hyperbolic cosine:
{\displaystyle \cosh x=\cos(ix)} \cosh x=\cos(ix)
Hyperbolic tangent:
{\displaystyle \tanh x=-i\tan(ix)} \tanh x=-i\tan(ix)
Hyperbolic cotangent:
{\displaystyle \coth x=i\cot(ix)} \coth x=i\cot(ix)
Hyperbolic secant:
{\displaystyle \operatorname {sech} x=\sec(ix)} \operatorname {sech} x=\sec(ix)
Hyperbolic cosecant:
{\displaystyle \operatorname {csch} x=i\csc(ix)} \operatorname {csch} x=i\csc(ix)
where i is the imaginary unit with the property that {\displaystyle i^{2}=-1.} {\displaystyle i^{2}=-1.}

The complex forms in the definitions above derive from Euler s formula.

Characterizing properties
Hyperbolic cosine
It can be shown that the area under the curve of cosh (x) over a finite interval is always equal to the arc length corresponding to that interval:[13]

{\displaystyle {\text{area}}=\int _{a}^{b}{\cosh {(x)}}\ dx=\int _{a}^{b}{\sqrt {1+\left({\frac {d}{dx}}\cosh {(x)}\right)^{2}}}\ dx={\text{arc length}}} {\text{area}}=\int _{a}^{b}{\cosh {(x)}}\ dx=\int _{a}^{b}{\sqrt {1+\left({\frac {d}{dx}}\cosh {(x)}\right)^{2}}}\ dx={\text{arc length}}

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