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




Question Number 129867 by Bird last updated on 20/Jan/21
calculate ∫   ((2x−1)/((x^2 −x+1)^3 ))dx
$${calculate}\:\int\:\:\:\frac{\mathrm{2}{x}−\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$
Answered by Olaf last updated on 20/Jan/21
Ω = ∫((2x−1)/((x^2 −x+1)^3 ))dx = ∫(du/u^3 )  (u = x^2 −x+1)  Ω = −(1/(2u^2 ))+C = −(1/(2(x^2 −x+1)^2 ))+C
$$\Omega\:=\:\int\frac{\mathrm{2}{x}−\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\:=\:\int\frac{{du}}{{u}^{\mathrm{3}} } \\ $$$$\left({u}\:=\:{x}^{\mathrm{2}} −{x}+\mathrm{1}\right) \\ $$$$\Omega\:=\:−\frac{\mathrm{1}}{\mathrm{2}{u}^{\mathrm{2}} }+\mathrm{C}\:=\:−\frac{\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }+\mathrm{C} \\ $$

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