Mathc initiation/Fichiers c : c77di4

Sommaire

Calculons la primitive :

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

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


        /                                                /       
       | csc(x)**n  dx = - csc(x)**(n-2) cot(x) - (n-2) | (csc(x)**n -     csc(x)**(n-2)) dx
       /                                                /
       

        /                                                /                        /
       | csc(x)**n  dx = - csc(x)**(n-2) cot(x) - (n-2) | (csc(x)**n dx +  (n-2) |  csc(x)**(n-2)) dx
       /                                                /                        /
       


        /                       /                        /
       | csc(x)**n  dx + (n-2) |  csc(x)**n dx =  (n-1) | csc(x)**n  dx 
       /                       /                        /
       
      
        /                                                 / 
 (n-1) | csc(x)**n  dx = - csc(x)**(n-2) cot(x)  + (n-2) | csc(x)**(n-2)) dx
       /                                                 /
                                                     
                      
       
        /                      1                             (n-2)   / 
       | csc(x)**n  dx = (-) -----  csc(x)**(n-2) cot(x)  +  -----  | csc(x)**(n-2)) dx
       /                     (n-1)                           (n-1)  /