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Question Number 188034 by Michaelfaraday last updated on 25/Feb/23

solve  ∫((x^2 +3)/(x^6 (x^2 +1)))dx

$${solve} \\ $$$$\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{6}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$

Answered by MJS_new last updated on 25/Feb/23

∫((x^2 +3)/(x^6 (x^2 +1)))dx=  =−2∫(dx/(x^2 +1))+2∫(dx/x^2 )−2∫(dx/x^4 )+3∫(dx/x^6 )=  =−2arctan x −(2/x)+(2/(3x^3 ))−(3/(5x^5 ))+C

$$\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{6}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx}= \\ $$$$=−\mathrm{2}\int\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{1}}+\mathrm{2}\int\frac{{dx}}{{x}^{\mathrm{2}} }−\mathrm{2}\int\frac{{dx}}{{x}^{\mathrm{4}} }+\mathrm{3}\int\frac{{dx}}{{x}^{\mathrm{6}} }= \\ $$$$=−\mathrm{2arctan}\:{x}\:−\frac{\mathrm{2}}{{x}}+\frac{\mathrm{2}}{\mathrm{3}{x}^{\mathrm{3}} }−\frac{\mathrm{3}}{\mathrm{5}{x}^{\mathrm{5}} }+{C} \\ $$

Commented by Michaelfaraday last updated on 01/Mar/23

thanks sir

$${thanks}\:{sir} \\ $$

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