Mathc initiation/Fichiers h : c80e
Vérifions cette égalité :
/ 1 1
| --------- dx = --- atanh(x/a) + c (a**2-x**2 => x**2 < a**2)
/ a**2-x**2 a
*****************
posons :
y = atanh(x/a) (*) y = atanh(x/a)
tanh(y) = tanh(atanh(x/a))
tanh(y) = (x/a) (**) tanh(y) = (x/a)
tanh(y)' = (x/a)'
1
y' sech(y)**2 = ---
a
1 1
y' = --- ---------
a sech(y)**2
1 1 sec(y)**2+tan(y)**2 = 1
y' = --- ----------- sec(y)**2 = 1-tan(y)**2
a 1-tan(y)**2
1 1 tan(y) = (x/a) (**) tanh(y) = (x/a)
y' = --- --------- tan(y)**2 = (x/a)**2
a 1-(x/a)**2
1 a**2 1
y' = --- ---- ---------
a a**2 1-(x**2/a**2)
1 a**2
y' = --- ---------
a a**2-x**2
a
y' = ---------
a**2-x**2
a
y' = ---------
a**2-x**2
/ / a
| y' dx = | --------- dx
/ / a**2-x**2
/ a
y = | --------- dx
/ a**2-x**2
(*) y = atanh(x/a)
/ a
atanh(x/a) = | --------- dx
/ a**2-x**2
1 / 1
--- atanh(x/a) = | --------- dx
a / a**2-x**2
/ 1 1
| --------- dx = --- atanh(x/a) + c égalité vérifié.
/ a**2-x**2 a