Answer
$$ - \frac{8}{3}\ln \left| {\sin \left( { - \frac{3}{8}x} \right)} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\cot \left( { - \frac{3}{8}x} \right)} dx \cr
& {\text{set }}u = - \frac{3}{8}x{\text{ then }}\frac{{du}}{{dx}} = - \frac{3}{8},\,\,\,\,\,\,\,\, - \frac{8}{3}du = dx \cr
& {\text{write the integrand in terms of }}u \cr
& \int {\cot \left( { - \frac{3}{8}x} \right)} dx = \int {\cot u} \left( { - \frac{8}{3}du} \right) \cr
& {\text{use multiple constant rule}} \cr
& = - \frac{8}{3}\int {\cot u} du \cr
& {\text{ using the Basic Trigonometric integral }}\int {\cot x} dx = \ln \left| {\sin x} \right| + C{\text{ }}\left( {{\text{see page 694}}} \right) \cr
& = - \frac{8}{3}\ln \left| {\sin u} \right| + C \cr
& {\text{write in terms of }}x \cr
& = - \frac{8}{3}\ln \left| {\sin \left( { - \frac{3}{8}x} \right)} \right| + C \cr} $$
