Answer
$\mathop \smallint \limits_0^\pi \left( {\sin \theta ,\theta ,\cos 2\theta } \right){\rm{d}}\theta = \left( {2,\frac{{{\pi ^2}}}{2},0} \right)$
Work Step by Step
$\mathop \smallint \limits_0^\pi \left( {\sin \theta ,\theta ,\cos 2\theta } \right){\rm{d}}\theta = \left( {\mathop \smallint \limits_0^\pi \sin \theta {\rm{d}}\theta ,\mathop \smallint \limits_0^\pi \theta {\rm{d}}\theta ,\mathop \smallint \limits_0^\pi \cos 2\theta {\rm{d}}\theta } \right)$
Evaluate $\mathop \smallint \limits_0^\pi \sin \theta {\rm{d}}\theta $
$\mathop \smallint \limits_0^\pi \sin \theta {\rm{d}}\theta = - \cos \theta |_0^\pi = 2$
Evaluate $\mathop \smallint \limits_0^\pi \theta {\rm{d}}\theta $
$\mathop \smallint \limits_0^\pi \theta {\rm{d}}\theta = \frac{1}{2}{\theta ^2}|_0^\pi = \frac{{{\pi ^2}}}{2}$
Evaluate $\mathop \smallint \limits_0^\pi \cos 2\theta {\rm{d}}\theta $
$\mathop \smallint \limits_0^\pi \cos 2\theta {\rm{d}}\theta = \frac{1}{2}\sin 2\theta |_0^\pi = 0$
Thus,
$\mathop \smallint \limits_0^\pi \left( {\sin \theta ,\theta ,\cos 2\theta } \right){\rm{d}}\theta = \left( {2,\frac{{{\pi ^2}}}{2},0} \right)$
