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   Integration of Hyperbolic Functions
The 6 basic hyperbolic functions are defined by
 hyperbolic sine hyperbolic cosine
 hyperbolic tangent hyperbolic cotangent
 hyperbolic secant hyperbolic cosecant
There are the following differentiation and integration formulas for hyperbolic functions:
  integral of the hyperbolic sine function
  integral of the hyperbolic cosine function
  integral of the hyperbolic secant squared function
  integral of the hyperbolic cosecant squared function
  
  
We provide here a list of useful hyperbolic identities:



  • When an integrand contains a hyperbolic function, the integral can be reduced to integrating a rational function by using the substitution .

       Example 1
    Calculate the integral .

    Solution.
    We make the substitution: u = 2 + 3sinh x, du = 3cosh xdx. Then . Hence, the integral is
         
       Example 2
    Evaluate .

    Solution.
    Since , and, hence, , we can write the integral as
         
    Making the substitution u = cosh x, du = sinh xdx, we obtain
         
       Example 3
    Evaluate the integral .

    Solution.
    We use integration by parts: . Let . Then, . Hence, the integral is
         
       Example 4
    Evaluate the integral .

    Solution.
    Since , we obtain
         
       Example 5
    Find the integral .

    Solution.
    By definition of the hyperbolic cosine, . Hence, the integral is equal
         
       Example 6
    Find the integral .

    Solution.
    By definition, and . Hence,
         
    We make the substitution: u = e x, du = e xdx and calculate the initial integral:
         
       Example 7
    Calculate the integral .

    Solution.
    Applying the formulas and , we get
         
       Example 8
    Evaluate the integral .

    Solution.
    We use integration by parts. Let
         
    Then
         
    Apply integration by parts again to the latter integral. Let
         
    Then we have
         
    Solving this equation for , we obtain the complete answer:
         

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