Mathc initiation/Fichiers c : c77di3

Sommaire

Calculons la primitive :

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

        u =       sec(x)**(n-2)                         dv =   sec(x)**2 dx
       du = (n-2) sec(x)**(n-3)  sec(x)tan(x) dx         v =   tan(x)   
       
       
           (udv)                     (u*v)               (v*du)
        /                                          /       
       | sec(x)**n  dx = sec(x)**(n-2) * tan(x) - | tan(x) * (n-2) sec(x)**(n-3)  sec(x) tan(x) dx
       /                                          /
            
            
        /                                                /       
       | sec(x)**n  dx = sec(x)**(n-2) * tan(x) - (n-2) | tan(x)**2       sec(x)**(n-2)  dx
       /                                                /
            
                                                                               tan(x)**2 = (sec(x)**2 - 1) 
                                                                                     
        /                                                /       
       | sec(x)**n  dx = sec(x)**(n-2) * tan(x) - (n-2) | (sec(x)**2 - 1)  sec(x)**(n-2) dx
       /                                                /

        /                                                /       
       | sec(x)**n  dx = sec(x)**(n-2) * tan(x) - (n-2) |  (sec(x)**n -   sec(x)**(n-2)) dx
       /                                                /


        /                                                 /                      / 
       | sec(x)**n  dx = sec(x)**(n-2) * tan(x) - (n-2)  |  sec(x)**n dx + (n-2) | sec(x)**(n-2) dx
       /                                                 /                       /
       


        /                       /                        /
       | sec(x)**n  dx + (n-2) |  sec(x)**n dx =  (n-1) | sec(x)**n  dx 
       /                       /                        /
       
      
        /                                                 / 
 (n-1) | sec(x)**n  dx = sec(x)**(n-2) * tan(x)  + (n-2) | sec(x)**(n-2) dx
       /                                                 /
                                                     
                      
       
        /                   1                               (n-2)   / 
       | sec(x)**n  dx =  -----  sec(x)**(n-2) * tan(x)  +  -----  | sec(x)**(n-2) dx
       /                  (n-1)                             (n-1)  /