Mathc initiation/Fichiers c : c77ce2


Sommaire


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

    Donc  
    
    sqrt(a**2-x**2) =   sqrt(a**2-(a sin(Ø))**2)               1           = cos(Ø)**2 + sin(Ø)**2 
                    = a sqrt(1-sin(Ø)**2)                      1-sin(Ø)**2 = cos(Ø)**2
                    = a sqrt(cos(Ø)**2)
                    = a      cos(Ø)


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