Integrating Powers of Cotangents and Cosecants

By Mark Zegarelli

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:

  1. Peel off a csc2 x and place it next to the dx:

    image0.png

  2. Use the trig identity 1 + cot2 x = csc2 x to express the remaining cosecant factors in terms of cotangents:

    image1.png

  3. Use the variable substitution u = cot x and du = –csc2 x dx:

    image2.png

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

image3.png

You can use trig identities to express it as cotangents and cosecants:

image4.png