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JS-dx-x-8-x-2-1-




Question Number 108325 by john santu last updated on 16/Aug/20
   ((⊸JS⊸)/(−−−−))  ∫ (dx/(x^8 (x^2 +1))) = ?
$$\:\:\:\frac{\multimap{JS}\multimap}{−−−−} \\ $$$$\int\:\frac{{dx}}{{x}^{\mathrm{8}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:=\:? \\ $$
Commented by john santu last updated on 16/Aug/20
   ((⊸JS⊸)/(______))  ∫ (dx/(x^8 (x^2 +1))) = ∫((1/(x^2 +1))−(1/x^2 )+(1/x^4 )−(1/x^6 )+(1/x^8 ))dx  = tan^(−1) (x)+(1/x)−(1/(3x^3 ))+(1/(5x^5 ))−(1/(7x^7 )) + c
$$\:\:\:\frac{\multimap{JS}\multimap}{\_\_\_\_\_\_} \\ $$$$\int\:\frac{{dx}}{{x}^{\mathrm{8}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:=\:\int\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }−\frac{\mathrm{1}}{{x}^{\mathrm{6}} }+\frac{\mathrm{1}}{{x}^{\mathrm{8}} }\right){dx} \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left({x}\right)+\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{\mathrm{3}{x}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{5}{x}^{\mathrm{5}} }−\frac{\mathrm{1}}{\mathrm{7}{x}^{\mathrm{7}} }\:+\:{c} \\ $$

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