The integral of
cot(x) is
ln∣sin(x)∣+C.
Proof. By definition, we have that
cot(x)=tan(x)1=sin(x)cos(x). So:
∫cot(x)dx=∫tan(x)1dx=∫sin(x)cos(x)dx.
Now we want to apply the substitution method. Let
u=sin(x). Then we get:
dxdu=cos(x)⟺du=cos(x)dx.
Together, we have that:
∫cot(x)dx=∫tan(x)1dx=∫sin(x)cos(x)dx=∫u1du=ln∣u∣+C=ln∣sin(x)∣+Csince u=sin(x).
Therefore, the integral of
cot(x) is
ln∣sin(x)∣+C.