Mathc initiation/Fichiers h : c80a
Vérifions cette égalité :
/ 1 1
| --------- dx = --- atan(x/a) + c
/ a**2+x**2 a
*****************
posons :
y = atan(x/a) (*)
tan(y) = tan(atan(x/a))
tan(y) = (x/a) (**)
tan(y)' = (x/a)'
1
y' sec(y)**2 = ---
a
1 1
y' = --- ---------
a sec(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) (**)
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 = atan(x/a) (*)
/ a
atan(x/a) = | --------- dx
/ a**2+x**2
1 / 1
--- atan(x/a) = | --------- dx
a / a**2+x**2
/ 1 1
| --------- dx = --- atan(x/a) + c égalité vérifié.
/ a**2+x**2 a