When evaluating the following integral, the integral with cotangent to the fifth power of XDX, we're going to pull out one of the cotangent X's and we're going to change all the rest into cosecants. So we get cotangent squared X ^2 DX. So the integral cotangent X times the quantity of cosecant squared X -, 1 ^2 DX. If we foil that out, we get cosecant to the 4th X -, 2, cosecant squared X + 1. That's still the integral times, the cotangent XDX. So now if I distribute that into three terms, we're going to get the integral of cotangent X, cosecant X. I'm pulling one of them out and leaving the cosecant cubed XDX because we know that the derivative of cosecant is negative cotangent cosecant. So then I'm going to do the same thing for the next term -2 cotangent X. I'm going to pull out one of the cosecant X That leaves a cosecant XDX. And then the last term is just positive integral of cotangent XDX. If I let AU sub so U equal cosec and X, so DU is negative cosec and X cotangent X DX, then we get negative the integral U cubed DU +2 the integral udu. Plus we know that the integral of cotangent X DX is just natural log of sine X + C. So when we evaluate these integrals, we get negative 1/4 U to the 4th plus U ^2 plus natural log sine X + C, sine natural log of absolute value of sine X + C. Substituting our U back end, we get negative 1/4 cosecant to the 4th X plus cosecant squared X plus natural log absolute value of sine X + C. And that would be an OK answer, except this is a multiple choice problem and that's not one of our answers. So what we're going to do is we're going to manipulate it so that we get them all in terms of cotangent X and see if that's one of our options. So instead of cosecant to the 4th X, think about that as one plus cotangent squared X ^2. And instead of cosecant squared X1 plus cotangent squared X. Then if we just do some distribution and combining like terms, we do see that this is one of our options. Negative 1/4 cotangent to the 4th X plus half cotangent squared X plus natural log of the absolute value sine X plus some new C because this old C would have been negative 1/4 + 1 plus the C. So this is a different constant than that C there.