سلسلة التكامل

\[{\color{Red} Q1})y^{12}=x-1\Rightarrow 12y^{11}dy=dx\\\int \frac{\sqrt[3]{x-1}+\sqrt[4]{x-1}}{(x-1)(1+\sqrt[6]{x-1})}=12\int \frac{y^{4}+{\color{Red} y^{2}}+y^{3}-{\color{Red} y^{2}}}{y\left ( 1+y^{2} \right )}dy\\=12\int ydy+12\int \frac{y^{2}-y}{y^{2}+1}dy=6y^{2}+12\int \frac{y^{2}+1-y-1}{y^{2}+1}dy\\=6y^{2}+12\int dy-12\int \frac{y}{y^{2}+1}dy-12\int \frac{1}{y^{2}+1}dy\\=6y^{2}+12y-6ln|y^{2}+1|-12tan^{-1}y+C\\=6\sqrt[6]{x-1}+12\sqrt[12]{x-1}-6ln|\sqrt[6]{x-1}+1|-12tan^{-1}\sqrt[6]{x-1}+C\]