Mathc initiation/a00aj

Sommaire

Calculons la primitive :

Calculer la primitive de 

       
   /              /   1         
  | csch(x)  dx = | --------  dx     
  /               /  sinh(x)                             
    
    
                   /     1          
                = |  --------     (1)  dx      
                  /   sinh(x)            

                  /            sinh(x)   
                = |  --------  ------- dx      
                 /   sinh(x)   sinh(x)                 
                                        
                                                 ________________________________
                   /    sinh(x)                  | cosh**2(x) - sinh**2(x) = 1  |
                = |  ----------- dx              |                              |
                  /   sinh**2(x)                 | -sinh**2(x)= -cosh**2(x) + 1 |
                                                 |                              |
                                                 |  sinh**2(x)=  cosh**2(x) - 1 |
                                                 |______________________________|
                                                       
                                                 ___________________
                   /     1                       |  u = cosh(x)    |    
                = | --------------- sinh(x) dx   | du = sinh(x) dx | 
                  /  cosh**2(x) - 1              |_________________|                                           
       

                                                       
                       /     1                     
                =     |  ----------  du          u**2 > 1 
                      /   u**2 - 1                  
                                                   
                                                   
                =      - acoth(u)       + c                        
      
                =      - acoth(cosh(x)) + c    
                
                
                                   ___________________________
                                   |  acoth(x) = atanh(1/x)  | 
                
                =      - acoth(u)  + c         =    -atanh(1/u)     + c              
      
                =      - acoth(cosh(x)) + c    = -atanh(1/cosh(x))  + c