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If-dx-x-3-1-x-6-2-3-xf-x-1-x-6-1-3-C-where-C-is-constant-of-integration-then-find-f-x-




Question Number 208791 by MATHEMATICSAM last updated on 23/Jun/24
If ∫ (dx/(x^3 (1 + x^6 )^(2/3) )) = xf(x).(1 + x^6 )^(1/3)  + C   where C is constant of integration then  find f(x).
$$\mathrm{If}\:\int\:\frac{{dx}}{{x}^{\mathrm{3}} \left(\mathrm{1}\:+\:{x}^{\mathrm{6}} \right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:=\:{xf}\left({x}\right).\left(\mathrm{1}\:+\:{x}^{\mathrm{6}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:{C}\: \\ $$$$\mathrm{where}\:{C}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{integration}\:\mathrm{then} \\ $$$$\mathrm{find}\:{f}\left({x}\right). \\ $$
Commented by mr W last updated on 23/Jun/24
∫...=−(((1+x^6 )^(1/3) )/(2x^2 ))+C  ⇒f(x)=−(1/(2x^3 ))
$$\int…=−\frac{\left(\mathrm{1}+{x}^{\mathrm{6}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} }{\mathrm{2}{x}^{\mathrm{2}} }+{C} \\ $$$$\Rightarrow{f}\left({x}\right)=−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{3}} } \\ $$

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