# dx-x-2-x-1-

Question Number 65988 by mmkkmm000m last updated on 07/Aug/19
$$\int{dx}/{x}^{\mathrm{2}} −{x}+\mathrm{1} \\$$
Commented by mathmax by abdo last updated on 07/Aug/19
$${let}\:{A}\:=\int\:\:\frac{{dx}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\:\Rightarrow{A}\:=\int\:\:\:\frac{{dx}}{\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \:+\frac{\mathrm{3}}{\mathrm{4}}} \\$$$$=_{{x}−\frac{\mathrm{1}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{t}} \:\frac{\mathrm{4}}{\mathrm{3}}\:\:\:\int\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{dt}\:=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}{arctan}\left(\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)+{c}\:. \\$$
Answered by meme last updated on 07/Aug/19
$$\:\:\:\:\:=\:\:\:−\int−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{dx}−{x}+\mathrm{1} \\$$$$\:\:\:\:\:\:\:=\:−\frac{\mathrm{1}}{{x}}\:−{x}+\mathrm{1} \\$$$$\:\:\:\:\:\:=\:−\frac{\mathrm{1}−{x}+{x}^{\mathrm{2}} }{{x}} \\$$$$\:\:\:\:\:\: \\$$
Answered by meme last updated on 07/Aug/19
$$\:\:\:\:\:\:\:\:{b}^{\mathrm{2}} −\mathrm{4}{ac}=\mathrm{1}−\mathrm{4}=−\mathrm{3}<\mathrm{0} \\$$$$\:\:\:\:\:\int\frac{{dx}}{\mathrm{1}−{x}+{x}^{\mathrm{2}} }=\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}{arctan}\left(\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)+{c} \\$$
Answered by AnjanDey last updated on 07/Aug/19
$$=\int\frac{{dx}}{{x}^{\mathrm{2}} −\mathrm{2}\centerdot{x}\centerdot\frac{\mathrm{1}}{\mathrm{2}}+\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} } \\$$$$=\int\frac{{dx}}{\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{\mathrm{2}} } \\$$$$=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{{x}−\frac{\mathrm{1}}{\mathrm{2}}}{\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}\right)+{C} \\$$$$=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)+{C} \\$$