[tex]t=\frac{\pi }{2}-x\Rightarrow dx=-dt[/tex]
[tex]\Rightarrow I=\int_{\frac{\pi }{3}}^{\frac{\pi }{6}}(\sqrt[3]{cost}-\sqrt[3]{sint})(-dt)=-\int_{\frac{\pi }{6}}^{\frac{\pi }{3}}(\sqrt[3]{sint}-\sqrt[3]{cost})dt=-I\Rightarrow 2I=0\Rightarrow I=0[/tex]