$y=\int _{-\pi }^{\pi }{{x}^{12}}{{\sin }^{9}}xdx……(1)$

Use king theorem of definite integral

$\int _{a}^{b}f(t)dt=\int _{a}^{b}f(a+b-t)dt$

$y=\int _{-\pi }^{\pi }{{\left( \pi -\pi -x \right)}^{12}}{{\sin }^{9}}\left( \pi -\pi -x \right)dx$

$y=\int _{-\pi }^{\pi }-{{x}^{12}}\sin 9xdx…..(2)$

Adding equation (1) and (2)

$2y=\int _{-\pi }^{\pi }{{x}^{12}}{{\sin }^{9}}dx+\left( -\int _{-\pi }^{\pi }{{x}^{12}}{{\sin }^{9}}xdx \right)$

$y=0$