Question Number 206537 by necx122 last updated on 18/Apr/24
$$\int\frac{{x}^{\mathrm{4}} −\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}}{dx} \\ $$
Answered by Berbere last updated on 18/Apr/24
$$=\int\frac{{x}\left(\mathrm{4}{x}^{\mathrm{3}} +\mathrm{2}{x}\right)−\mathrm{2}{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{2}{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$=\int\frac{{x}.\left(\sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\right)'}{{x}^{\mathrm{2}} }−\frac{\mathrm{1}.\sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}}{{x}^{\mathrm{2}} }{dx} \\ $$$$=\int{d}\left(\frac{\sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}}{{x}}\right)=\frac{\sqrt{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}}{{x}}+{a};{a}\in\mathbb{R} \\ $$$$\: \\ $$