Mathc initiation/a00aj
Calculons la primitive :
Calculer la primitive de
/ / 1
| csch(x) dx = | -------- dx
/ / sinh(x)
/ 1
= | -------- (1) dx
/ sinh(x)
/ sinh(x)
= | -------- ------- dx
/ sinh(x) sinh(x)
________________________________
/ sinh(x) | cosh**2(x) - sinh**2(x) = 1 |
= | ----------- dx | |
/ sinh**2(x) | -sinh**2(x)= -cosh**2(x) + 1 |
| |
| sinh**2(x)= cosh**2(x) - 1 |
|______________________________|
___________________
/ 1 | u = cosh(x) |
= | --------------- sinh(x) dx | du = sinh(x) dx |
/ cosh**2(x) - 1 |_________________|
/ 1
= | ---------- du u**2 > 1
/ u**2 - 1
= - acoth(u) + c
= - acoth(cosh(x)) + c
___________________________
| acoth(x) = atanh(1/x) |
= - acoth(u) + c = -atanh(1/u) + c
= - acoth(cosh(x)) + c = -atanh(1/cosh(x)) + c