Chapter 7 - Section 7.1 - Integration by Parts - 7.1 Exercises - Page 490: 17

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

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