Mathc initiation/Fichiers c : c74c09


Sommaire


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c01i.c
/* ---------------------------------- */
/* save as c1i.c                      */
/* ---------------------------------- */
#include "x_hfile.h"
#include      "fi.h"
/* ---------------------------------- */
int main(void)
{
int      n =  2*50;
double   a =  .5;
double   b =   1;

 clrscrn();

 printf(" With the Simpson's rule.    (n = %d)\n\n"
        "    (%.3f\n"
        " int(      (%s)  dx = %.6f\n"
        "    (%.3f\n\n\n\n",n,  b, feq, simpson(f,a,b,n), a);

 printf(" With the antiderivative of f.\n\n"
        " F(x) = %s \n\n\n" 
        " F(%.3f) -  F(%.3f)  = %.6f \n\n\n", Feq, b,a, F(b)-F(a));
 
 stop();

 return 0;
}
/* ---------------------------------- */


Calculons l'intégrale avec la fonction simpson(f,a,b,n); puis avec sa primitive F(x).


Exemple de sortie écran :
 With the Simpson's rule.    (n = 100)

    (1.000
 int(      (sech(x))  dx = 0.385388
    (0.500



 With the antiderivative of f.

 F(x) = atan(sinh(x)) 


 F(1.000) -  F(0.500)  = 0.385388 


 Press return to continue.



Calculons la primitive :
             
Calculer la primitive de 

       
   /              /   1         
  | sech(x)  dx = | --------  dx     
  /               /  cosh(x)                             
    
    
                   /     1          
                = |  --------     (1)  dx      
                  /   cosh(x)            

                  /            cosh(x)   
                = |  --------  ------- dx      
                 /   cosh(x)   cosh(x)                 
                                        
                                                 ______________________________
                   /    cosh(x)                  | cosh**2(x) - sinh**2(x) = 1|
                = |  ----------- dx              |                            |
                  /   cosh**2(x)                 | cosh**2(x) = sinh**2(x) + 1|
                                                 |____________________________|
                                                       
                                                  ___________________
                   /     1                       |  u = sinh(x)    |    
                = | --------------- cosh(x) dx   | du = cosh(x) dx | 
                  /  sinh**2(x) + 1              |_________________|                                           
       

                                                       
                       /     1                     
                =     |  ----------  du            
                      /   u**2 + 1                  

      
                =       atan(u)  + c
      
                =       atan(sinh(x))  + c

Remarque : (atan(u))' = 1/(u**2 + 1)

       (atan(u))'     =     1/(u**2 + 1)

      /                   /
     | (atan(u))' du  =   | 1/(u**2 + 1) du
     /                    /


                          /
        atan(u)       =  |  1/(u**2 + 1) du
                         /

   /
  |  1/(u**2 + 1) du  =  atan(u) + c
  /