Integral

\hspace{-20}\bf{(a)::}$Given $\bf{\int\frac{1}{\left(\sin^{11}(x)\cdot \cos (x)\right)^{\frac{1}{3}}}dx}$\\\\\\ $\bf{\displaystyle =\int\frac{1}{\left(\sin^{12}(x)\cdot \cos (x)\cdot \frac{1}{\sin (x)}\right)^{\frac{1}{3}}}dx}$\\\\\\ $\bf{=\int\frac{1}{\sin^4(x)\cdot \cot^{\frac{1}{3}}(x)}dx = \int\frac{\csc^2(x)\cdot(1+\cot^2(x))}{\cot^{\frac{1}{3}}(x)}dx}$\\\\\\ Let $\bf{\cot(x)=t^3,}$ Then $\bf{\csc^2(x)dx=-3t^2dt}$\\\\\\ $\bf{=-3\int\frac{t^2\cdot(1+t^6)}{t}dt = -3\int (t+t^7)dt = -3\left\{\frac{t^2}{2}+\frac{t^8}{8}\right\}+\mathbb{C}}$\\\\\\ $\bf{=-3\left\{\frac{\cot^{\frac{2}{3}}(x)}{2}+\frac{\cot^{\frac{8}{3}}(x)}{8}\right\}+\mathbb{C}}$

\hspace{-20}\bf{(b)::}$Given $\bf{\int \sqrt{\frac{\cos^3 (x)}{\sin^{11}(x)}}dx = \int\sqrt{\frac{\cos^3(x)}{\sin^3(x)}\cdot \frac{1}{\sin^8(x)}}dx}$\\\\\\ $\bf{=\int \frac{1}{\sin^{4}(x)}\cdot \left(\cot (x)\right)^{\frac{3}{2}}dx = \int \csc^2(x)\cdot (1+\cot^2(x))\cdot \left(\cot (x)\right)^{\frac{3}{2}}dx}$\\\\\\ Now Let $\bf{\cot (x)=t^2\;,}$ Then $\bf{\csc^2(x)dx = -2tdt}$\\\\\\ So Integral is $\bf{=-2\int (1+t^4)\cdot t^3dt = -2\int \left(t^3+t^7\right)dt}$\\\\\\ $\bf{=-2\left\{\frac{t^4}{4}+\frac{t^8}{8}\right\}+\mathbb{C} = -2\left\{\frac{\cot^2(x)}{4}+\frac{\cot^4(x)}{8}\right\}+\mathbb{C}}$