Mathc initiation/Fichiers c : c77di2

Sommaire

Calculons la primitive :

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

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


   /                                                /       
  | cos(x)**n  dx = cos(x)**(n-1) * sin(x) + (n-1) |  (cos(x)**(n-2) -          cos(x)**n)   dx
  /                                                /


   /                                                /                           / 
  | cos(x)**n  dx = cos(x)**(n-1) * sin(x) + (n-1) |  cos(x)**(n-2) dx - (n-1) | cos(x)**n   dx
  /                                                /                           /
       

   /                                                /                       /                 /
  | cos(x)**n  dx = cos(x)**(n-1) * sin(x) + (n-1) |  cos(x)**(n-2) dx - n | cos(x)**n  dx + | cos(x)**n dx
  /                                                /                       /                 /
       
              
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n | cos(x)**n  dx = cos(x)**(n-1) * sin(x) + (n-1) |  cos(x)**(n-2) dx
  /                                                /                             
       
       
   /                 1                           (n-1)  /                       
  | cos(x)**n  dx = --- cos(x)**(n-1) * sin(x) + ----- |  cos(x)**(n-2) dx
  /                  n                             n   /