Mathc initiation/Fichiers c : c77di1

Sommaire

Calculons la primitive :

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

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


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


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


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