Mathc initiation/Fichiers h : c80d
Apparence
Vérifions cette égalité :
/ 1
| --------------- dx = asinh(x/a) + c
/ sqrt(a**2+x**2)
*****************
y = asinh(x/a) * y = asinh(x/a)
sinh(y) = sinh(asin(x/a))
sinh(y) = (x/a)
(sinh(y))' = (x/a)'
y' cosh(y) = (1/a)
1 1
y' = --- ------
a cosh(y)
1 1
y' = --- --------------- (-)sinh(y)**2+cosh(y)**2=1
a sqrt(cosh(y)**2) cosh(y)**2=1+sinh(y)**2
1 1
y' = --- ---------------
a sqrt(1+sinh(y)**2)
1 1 * y = asinh(x/a)
y' = --- -------------------------
a sqrt(1+sinh(asinh(x/a))**2)
sinh(asinh(x/a)) = (x/a)
1 1 sinh(asinh(x/a))**2 = (x/a)**2
y' = --- ----------------
a sqrt(1+(x/a)**2)
1 1 sqrt(a**2) = a
y' = ---------- ----------------
sqrt(a**2) sqrt(1+(x**2/a**2)) sqrt(a)sqrt(b)= sqrt(ab)
1
y' = ----------------
sqrt(a**2+x**2)
/ / 1
| y' dy = | ---------------- dx
/ / sqrt(a**2+x**2)
/ 1
y = | ---------------- dx
/ sqrt(a**2+x**2)
* y = asinh(x/a)
/ 1
asin(x/a) = | ---------------- dx
/ sqrt(a**2+x**2)
/ 1
| ---------------- dx = asinh(x/a) + c égalité vérifié.
/ sqrt(a**2+x**2)