Mathc initiation/Fichiers c : c72c09
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c01c.c |
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/* ---------------------------------- */
/* save as c1g.c */
/* ---------------------------------- */
#include "x_hfile.h"
#include "fg.h"
/* ---------------------------------- */
int main(void)
{
int n = 2*50;
double a = 2.;
double b = 3.;
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)
(3.000
int( (exp(3*x)/(exp(x)+1)) dx = 162.640500
(2.000
With the antiderivative of f.
F(x) = 1/2 exp(x) (exp(x)-2) + ln(exp(x)+1)
F(3.000) - F(2.000) = 162.640500
Press return to continue.
Calculons la primitive :
/
Calculer la primitive de | e**(3*x)/(e**x+1) dx
/ e**(3x) / e**x
| -------- dx = | e**(2x) [ ------ ] dx
/ e**x+1 / e**x+1
/ e**x +1-1 (+0 = +1-1)
= | e**(2x) [ --------- ] dx
/ e**x+1
/ e**x+1 1
= | e**(2x) [ ------ - ------ ] dx
/ e**x+1 e**x+1
/ 1
= | e**(2x) [ 1 - ------] dx
/ e**x+1
/ e**x
= | e**x [ e**x - ------] dx
/ e**x+1
/ e**x +1-1 (+0 = +1-1)
= | e**x [ e**x - --------- ] dx
/ e**x+1
/ ( e**x+1 1 )
= | e**x [ e**x -( ------ - ------ ) ] dx
/ ( e**x+1 e**x+1 )
/ 1
= | e**x [ e**x - 1 + ------ ] dx
/ e**x+1
/ e**x
= | [ e**(2x) - e**x + ------ ] dx
/ e**x+1
/ / / e**x
= | e**(2x) dx - | e**x dx + | ------ dx
/ / / e**x+1
___________________
| u = e**x+1 |
| du = e**x dx |
|___________________|
e**(2x) / 1
= ------- - e**x + | --- du
2 / u
e**(2x)
= ------- - e**x + ln(u) + c
2
e**(2x)
= ------- - e**x + ln(e**x+1) + c
2
e**(2x)-2 e**x
= --------------- + ln(e**x+1) + c
2
1
= --- e**x (e**x-2) + ln(e**x+1) + c
2