Integrating Powers of Cotangents and Cosecants
You can integrate powers of cotangents and cosecants similar to the way you do tangents and secant. For example, here’s how to integrate cot^{8} x csc^{6} x:

Peel off a csc^{2} x and place it next to the dx:

Use the trig identity 1 + cot^{2 }x = csc^{2} x to express the remaining cosecant factors in terms of cotangents:

Use the variable substitution u = cot x and du = –csc^{2} x dx:
At this point, the integral is a polynomial, and you can evaluate it.
Sometimes, knowing how to integrate cotangents and cosecants can be useful for integrating negative powers of other trig functions — that is, powers of trig functions in the denominator of a fraction.
For example, suppose that you want to integrate
You can use trig identities to express it as cotangents and cosecants: