Mathc initiation/Fichiers c : c77ce1


Sommaire


Etude de sqrt(a**2+x**2)
  
    Avec sqrt(a**2+x**2) posons x = a tan(Ø)

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


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