Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 516: 13

$-t \cot t + \ln |\sin t| + C$

Let $u = t, dv = \csc^2 t dt$ and $du = dt, v = - \cot t$ $\int t \csc^2 t dt$ $= -t \cot t - \int -\cot t dt$ $= -t \cot t + \int \frac{\cos t}{\sin t} dt$ $= -t \cot t + \int \frac{1}{z} dz$ $= -t \cot t + \ln |z| + C$ $= -t \cot t + \ln |\sin t| +C$