Answer
$= - t\cot t + \ln |\sin t| +C$
Work Step by Step
$\int t (\csc^{2}t) dx$
$u = t$
$u' = 1$
$\frac{du}{dt} = 1$
$du = 1 dt$
$du = dt$
$dv = (\csc^{2}t)$
$v = - \cot t$
$uv - \int vdu$
$= (t)(- \cot t) - \int (- \cot t)dt$
$= (t)(- \cot t) + \int (\cot t)dt$
$= (t)(- \cot t) + \ln |\sin t| +C$
$= - t\cot t + \ln |\sin t| +C$