Question

\(\int\dfrac{1}{\sqrt[3]{x^2}-\sqrt[4]{x^3}}\)

Answer 1

Đặt \(\sqrt[12]{x}=t\Rightarrow x=t^{12}\Rightarrow dx=12t^{11}dt\)

\(I=\int\dfrac{12t^{11}dt}{t^8-t^9}=\int\dfrac{12t^{11}dt}{t^8\left(1-t\right)}=\int\dfrac{12t^3dt}{1-t}\)

\(=-12\int\left(t^2+t+1+\dfrac{1}{t-1}\right)dt\)

\(=-12\left(\dfrac{t^3}{3}+\dfrac{t^2}{2}+t+ln\left|t-1\right|\right)+C\)

\(=-12\sqrt[4]{x}-6\sqrt[6]{x}-12\sqrt[12]{x}-12ln\left|\sqrt[12]{x}-1\right|+C\)