Integrate cot^2x

To integrate cot^2x, also written as ∫cot²x dx, cot squared x, (cot x)^2, and cot^2(x)we start by using standard trig identities to simplify the integral to a form we can work with.

We start with this standard and well-known trig identity for cot²x.

We rearrange the Pythagorean for cos²x so that we can substitute it into our previous trig identity.

Hence we now have this new expression for cot²x.

We can now expand the fraction, and notice that the sin²x terms cancel out leaving 1.

Hence we now have this new expression for cot²x.

Hence our integration problem can be rewritten to a different form which means the same thing. Although we would be able to integrate the 1 very easily to give x, the first term with sin²x will require more thinking.

We remember that cosecx = 1/sinx, which is a standard trig identity found in formula booklets. This is that annoying one where people write it as cosecant(x) and even csc(x). As you can see, if we square both sides, then the RHS looks like the first term of our integration problem.

We now rewrite our original integration problem in a new form, which means the same thing, as explained in earlier steps.

We recall from standard formula booklets that when we differentiate cotx we get -cosex²x, and this looks very much like the first term in our integration problem, except that it has a negative sign. However we can play a little trick with the signs as shown below.

If we were to multiply -1 with -cosec²x then it remains the same as +cosec²x.

Therefore, logic dictates that if the differential of cotx was -cosec²x, then the integral of -cosec²x should be cotx.

Finally, we are able to complete our integration problem. We remember that -1 behind the integral. The lone x comes from integrating the constant 1. We also introduce the +C which is the integration constant.