Answer
$$0$$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /\omega }^{\pi /\omega } {t\sin \left( {k\omega t} \right)} dt \cr
& {\text{Integrate by parts}} \cr
& u = t{\text{ }} \Rightarrow {\text{ }}du = dt \cr
& dv = \sin \left( {k\omega t} \right)dt{\text{ }} \Rightarrow {\text{ }}v = - \frac{1}{{k\omega }}\cos \left( {k\omega t} \right) \cr
& \int {t\sin \left( {k\omega t} \right)} dt = - \frac{t}{{k\omega }}\cos \left( {k\omega t} \right) + \int {\frac{1}{{k\omega }}\cos \left( {k\omega t} \right)} dt \cr
& {\text{ }} = - \frac{t}{{k\omega }}\cos \left( {k\omega t} \right) + \frac{1}{{{{\left( {k\omega } \right)}^2}}}\sin \left( {k\omega t} \right) + C \cr
& {\text{Therefore,}} \cr
& \int_{ - \pi /\omega }^{\pi /\omega } {t\sin \left( {k\omega t} \right)} dt = \left[ { - \frac{t}{{k\omega }}\cos \left( {k\omega t} \right) + \frac{1}{{{{\left( {k\omega } \right)}^2}}}\sin \left( {k\omega t} \right)} \right]_{ - \pi /\omega }^{\pi /\omega } \cr
& {\text{Evaluating}} \cr
& = \left[ { - \frac{t}{{k\omega }}\cos \left( {k\omega \left( {\frac{\pi }{\omega }} \right)} \right) + \frac{1}{{{{\left( {k\omega } \right)}^2}}}\sin \left( {k\omega \left( {\frac{\pi }{\omega }} \right)} \right)} \right] \cr
& - \left[ { - \frac{t}{{k\omega }}\cos \left( {k\omega \left( { - \frac{\pi }{\omega }} \right)} \right) + \frac{1}{{{{\left( {k\omega } \right)}^2}}}\sin \left( {k\omega \left( { - \frac{\pi }{\omega }} \right)} \right)} \right] \cr
& {\text{Simplifying}} \cr
& = \left[ { - \frac{t}{{k\omega }}\cos \left( {k\pi } \right) + \frac{1}{{{{\left( {k\omega } \right)}^2}}}\sin \left( {k\pi } \right)} \right] \cr
& - \left[ { - \frac{t}{{k\omega }}\cos \left( { - \pi k} \right) + \frac{1}{{{{\left( {k\omega } \right)}^2}}}\sin \left( {k\omega \left( { - \pi k} \right)} \right)} \right] \cr
& {\text{Then}} \cr
& = \left[ { - \frac{t}{{k\omega }}\cos \left( {k\pi } \right)} \right] - \left[ { - \frac{t}{{k\omega }}\cos \left( {\pi k} \right)} \right] \cr
& = - \frac{t}{{k\omega }}\cos \left( {k\pi } \right) + \frac{t}{{k\omega }}\cos \left( {\pi k} \right) \cr
& = 0 \cr} $$
