Answer
$$\frac{1}{6}\ln \left| {\sin {x^6}} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {{x^5}\cot {x^6}} dx \cr
& {\text{set }}u = {x^6}{\text{ then }}\frac{{du}}{{dx}} = 6{x^5},\,\,\,\,\,\,\,\,\frac{{du}}{{6{x^5}}} = dx \cr
& {\text{write the integrand in terms of }}u \cr
& \int {{x^5}\cot {x^6}} dx = \int {{x^5}\cot u} \left( {\frac{{du}}{{6{x^5}}}} \right) \cr
& {\text{cancel the common factor}} \cr
& = \int {\cot u} \left( {\frac{{du}}{6}} \right) \cr
& {\text{use multiple constant rule}} \cr
& = \frac{1}{6}\int {\cot u} du \cr
& {\text{ using the Basic Trigonometric integral }}\int {\cot x} dx = \ln \left| {\sin x} \right| + C{\text{ }}\left( {{\text{see page 694}}} \right) \cr
& = \frac{1}{6}\ln \left| {\sin u} \right| + C \cr
& {\text{write in terms of }}x \cr
& = \frac{1}{6}\ln \left| {\sin {x^6}} \right| + C \cr} $$
