You can integrate powers of cotangents and cosecants similar to the way you do tangents and secant. For example, here’s how to integrate cot8 x csc6 x:
Peel off a csc2 x and place it next to the dx:
Use the trig identity 1 + cot2 x = csc2 x to express the remaining cosecant factors in terms of cotangents:
Use the variable substitution u = cot x and du = –csc2 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:
