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Question Number 165615 by cortano1 last updated on 05/Feb/22

$${\int }_0^1\frac{dx}{\sqrt{x}(\sqrt[3]{x}+\sqrt[4]{x})}=?$$

Answered by Ar Brandon last updated on 05/Feb/22

$$I={\int }_0^1\frac{dx}{\sqrt{x}(\sqrt[3]{x}+\sqrt[4]{x})},x=t^{12}\Rightarrow dx=12t^{11}dt$$$$=12{\int }_0^1\frac{t^{11}}{t^6(t^4+t^3)}=12{\int }_0^1\frac{t^2}{t+1}dt=12{\int }_0^1\frac{{(t+1-1)}^2}{t+1}dt$$$$=12{\int }_0^1(t+1-2+\frac{1}{t+1})dt=12{[\frac{t^2}{2}-t+\ln (t+1)]}_0^1$$$$=12(\frac{1}{2}-1+\ln \left(2\right))=6(\ln 4-1)$$

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