\int\dfrac{\sin x}{\sqrt[3]{\cos^{2}x}}\mathrm{d}x=\int\sin x\cdot\left(\cos x\right)^{-\frac{2}{3}}\mathrm{d}x=-\int-\sin x\cdot\left(\cos x\right)^{-\frac{2}{3}}\mathrm{d}x=-\dfrac{\left(\cos x\right)^{\frac{1}{3}}}{\frac{1}{3}}=-3\cdot\sqrt[3]{\cos x}+C