Mathc initiation/Fichiers c : c77ce3


Sommaire


Etude de sqrt(x**2-a**2)
  
    Avec sqrt(x**2-a**2) posons x = a sec(Ø)

    Donc  
    
    sqrt(x**2-a**2) =   sqrt((a sec(Ø))**2-a**2)                  sec(Ø)**2 - tan(Ø)**2 = 1 
                    = a sqrt(sec(Ø)**2-1))                        sec(Ø)**2 -    1      = tan(Ø)**2
                    = a sqrt(tan(Ø)**2)
                    = a      tan(Ø)


                /
    Exemple :  | x/(sqrt(x**2-2**2) dx
               /
                                                         a = 2
                                                         x = 2 sec(Ø) 
                                                        dx = 2 tan(Ø) sec(Ø) 
                                                        
      /                              /
     |        x                     |        2sec(Ø)            
     | ------------------  dx =     |  ---------------------   2 tan(Ø) sec(Ø)  =
     |   sqrt(x**2-4)               |   sqrt((2sec(Ø))**2-4)
     /                              /
                                                                                                                                        
                                                        
                                     /
                                    |       2sec(Ø)             
                                    |  ---------------------   2 tan(Ø) sec(Ø)  =
                                    |   2  sqrt(sec(Ø)**2-1)
                                    /                                                        
                                                        
                                                        
                                     /
                                    |          sec(Ø)            
                                    |  ---------------------   2 tan(Ø) sec(Ø)  =
                                    |      sqrt(tan(Ø)**2) 
                                    /                                                        
                                                                                                                                                                    
                                                        
                                     /
                                    |          sec(Ø)            
                                    |  ---------------------   2 tan(Ø) sec(Ø)  =
                                    |          tan(Ø) 
                                    /                                                              
                                                        
                                                        
                                     /          
                                    |    2 sec(Ø)**2     =
                                    /       
                                                                                          
                                         
                                         2 tan(Ø) + c  =  
                                        
                                              
                  .               nous avons                  
                . .                        
              .   .                   x = 2   sec(Ø)                
         x  .     .  S                x = 2 (1/cos(Ø))                 
          .       .              cos(Ø) = 2/x  
        . . . . . .                                
             2                           
                                        
                                        
                  .                      
                . .                (2)**2 + S**2 =      x**2           
              .   .                         S**2 =      x**2-4    
         x  .     . sqrt(x**2-4)            S    = sqrt(x**2-4)          
          .       .                    
        . . . . . .                                            
            2                  
                            donc    2 tan(x) + c = 2 sqrt(x**2-4)/2 + c                tan(Ø) =  sqrt(x**2-4)/2 
                            
                            
                              
                             /
                            | x/(sqrt(x**2-4) dx = sqrt(x**2-4)  + c
                            /