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x-x-2-1-3-dx-




Question Number 204598 by SEKRET last updated on 22/Feb/24
        ∫((x−x^2 ))^(1/3)  dx
$$\:\:\:\:\:\:\:\:\int\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}} \\ $$$$ \\ $$
Commented by Frix last updated on 23/Feb/24
t=sin^(−1)  (√x) ⇒  (1/( (4)^(1/3) ))∫sin^(5/3)  2t dt  We need the Hypergeometric Function...
$${t}=\mathrm{sin}^{−\mathrm{1}} \:\sqrt{{x}}\:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{4}}}\int\mathrm{sin}^{\frac{\mathrm{5}}{\mathrm{3}}} \:\mathrm{2}{t}\:{dt} \\ $$$$\mathrm{We}\:\mathrm{need}\:\mathrm{the}\:\mathrm{Hypergeometric}\:\mathrm{Function}… \\ $$
Commented by SEKRET last updated on 24/Feb/24
 thank  you sir
$$\:\boldsymbol{\mathrm{thank}}\:\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{sir}} \\ $$
Answered by TonyCWX08 last updated on 23/Feb/24
$$ \\ $$
Commented by SEKRET last updated on 23/Feb/24
  ?   there is nothing
$$\:\:?\: \\ $$there is nothing

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