Mathc initiation/Fichiers c : c78di2

Sommaire

Calculons la primitive :

                            /
 Calculer la primitive de  | cosh(x)**n dx 
                           /
       
       Utilisons l'intégration par partie
       
        u = ...    dv = ...
       du = ...     v = ...
       
        /              /
       | u dv = u v - | v du
       /              /
             
       
        /               
       | cosh(x)**n  dx = 
       /               
       

        u =       cosh(x)**(n-1)                   dv = cosh(x) dx
       du = (n-1) cosh(x)**(n-2) sinh(x) dx         v = sinh(x)   
       
       
       (u dv)                   (u*v)               (v*du)
   /                                             /       
  | cosh(x)**n  dx = cosh(x)**(n-1) * sinh(x) - | sinh(x) * (n-1) cos(x)**(n-2) sin(x) dx
  /                                             /
            
            
   /                                                /       
  | cosh(x)**n  dx = cosh(x)**(n-1) sinh(x) - (n-1) | sinh(x)**2        cos(x)**(n-2) dx
  /                                                /
  
                                                                            sinh(x)**2 = (cosh(x)**2 - 1) 
                                                                                          
   /                                                 /       
  | cosh(x)**n  dx = cosh(x)**(n-1) sinh(x) - (n-1) | (cosh(x)**2 - 1)  cosh(x)**(n-2) dx
  /                                                 /


   /                                                 /       
  | cosh(x)**n  dx = cosh(x)**(n-1) sinh(x) - (n-1) |  (cosh(x)**n    - cosh(x)**(n-2)) dx
  /                                                 /


   /                                                 /                        / 
  | cosh(x)**n  dx = cosh(x)**(n-1) sinh(x) - (n-1) |  cosh(x)**n dx + (n-1) |  cosh(x)**(n-2) dx
  /                                                 /                        /
       
              
   /                                                                          /                       
n | cosh(x)**n  dx = cosh(x)**(n-1) sinh(x)                          + (n-1) |  cosh(x)**(n-2) dx
  /                                                                          /                             
       
       
   /                 1                            (n-1)  /                       
  | cosh(x)**n  dx = --- cosh(x)**(n-1) sinh(x) + ----- |  cosh(x)**(n-2) dx
  /                  n                              n   /