Đặt $t=\dfrac{\pi}{2}-x \Rightarrow dx=-dt$
$I=-\int_{\frac{\pi}{2}}^0 \bigg (\sqrt[3]{\cos ( \dfrac{\pi}{2}-t)} -\sqrt[3]{\sin (\dfrac{\pi}{2}-t)} \bigg )dt =\int_0^{\frac{\pi}{2}} \bigg ( \sqrt[3]{\sin t} -\sqrt[3]{\cos t} \bigg )dt$
$=-\int_0^{\frac{\pi}{2}} \bigg ( \sqrt[3]{\cos x} -\sqrt[3]{\sin x} \bigg ) dx=-I$
$\Rightarrow 2I = 0 \Rightarrow I=0$