Mathc initiation/a00ai

Sommaire

Calculons la primitive :

             
Calculer la primitive de 

       
   /              /   1         
  | sech(x)  dx = | --------  dx     
  /               /  cosh(x)                             
    
    
                   /     1          
                = |  --------     (1)  dx      
                  /   cosh(x)            

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

                                                       
                       /     1                     
                =     |  ----------  du            
                      /   u**2 + 1                  

      
                =       atan(u)  + c
      
                =       atan(sinh(x))  + c

Remarque : (atan(u))' = 1/(u**2 + 1)

       (atan(u))'     =     1/(u**2 + 1)

      /                   /
     | (atan(u))' du  =   | 1/(u**2 + 1) du
     /                    /


                          /
        atan(u)       =  |  1/(u**2 + 1) du
                         /

   /
  |  1/(u**2 + 1) du  =  atan(u) + c
  /