Calculus : int(cot^3x)dx but why

[kubigiri]

∫cot^3(x) dx

= ∫cot^2(x) cot (x) dx

= ∫(csc^2(x) - 1) cot (x) dx

= ∫ csc^2(x)) cot (x) dx - ∫cot (x) dx

= ∫- cot (x) d (cot x) - ln I sin x I

= - (1/2)cot^2(x) - ln I sin x I + C
but
if∫cot^3(x) dx

= ∫cot^2(x) cot (x) dx

= ∫(csc^2(x) - 1) cot (x) dx

= ∫ csc^2(x)) cot (x) dx - ∫cot (x) dx

= ∫- csc (x) d (csc x) - ln I sin x I

= - (1/2)csc^2(x) - ln I sin x I + CSo
= - (1/2)cot^2(x) + C = = - (1/2)csc^2(x) + C
csc = cot
but
csc^2 = cot^2 +1

i'm not sure but maybe "1" will add in C ? Please help me

 

[Orodruin]

In general, there is no guarantee that you will get the same integration constant so the 1 can be absorbed into the corresponding constant.

The check should be to differentiate the result to make sure you get the integrand back.