Mathc initiation/Fichiers c : c77ce2
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) dØ
/ /
| x | 2sin(Ø)
| ------------------ dx = | --------------------- 2 cos(Ø) dØ =
| sqrt(4-x**2) | sqrt(4-(2sin(Ø))**2)
/ /
/
| 2sin(Ø)
| --------------------- 2 cos(Ø) dØ =
| 2 sqrt(1-sin(Ø)**2)
/
/
| sin(Ø)
| --------------------- 2 cos(Ø) dØ =
| sqrt(cos(Ø)**2)
/
/
| sin(Ø)
| 2 ------ cos(Ø) dØ =
| cos(Ø)
/
/
| 2 sin(Ø) dØ =
/
-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
/