What is the integral of cosx/ sinx​

1 Use Integration by Substitution.
Let
u
=
cos
⁡
x
u=cosx,
d
u
=
−
sin
⁡
x
d
x
du=−sinxdx

2 Using
u
u and
d
u
du above, rewrite
∫
cos
⁡
x
sin
⁡
x
d
x
∫cosxsinxdx.
∫
−
u
d
u
∫−udu

3 Use Power Rule:
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
∫x
n
dx=
n+1
x
n+1

​
+C.
−
u
2
2
−
2
u
2

​

4 Substitute
u
=
cos
⁡
x
u=cosx back into the original integral.
−
2
x
2
−
2
cos
2
x
​

5 Add constant.
−
2
x
2
+
C
−
2
cos
2
x
​
+C

Done