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Question Number 121519 by benjo_mathlover last updated on 09/Nov/20

∫ ((1−x^4 )/(x^2 (√(x^4 +x^2 +1)))) dx ?

$$\:\:\int\:\frac{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }{\mathrm{x}^{\mathrm{2}} \:\sqrt{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}}}\:\mathrm{dx}\:? \\ $$

Answered by liberty last updated on 09/Nov/20

∫ ((1−x^4 )/(x^3 (√(x^2 +1+x^(−2) )))) dx = ∫ ((x^(−3) −x)/( (√(x^2 +x^(−2) +1)))) dx let v = x^2 +x^(−2) +1 →dv = 2x−2x^(−3) dx dv = −2(x^(−3) −x)dx ∫ ((−(1/2)dv)/( (√v))) = −(1/2)∫ v^(−(1/2)) dv = −(√v) + c = −(√(x^2 +x^(−2) +1)) + c = −((√(x^4 +x^2 +1))/x) + c

$$\:\int\:\frac{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }{\mathrm{x}^{\mathrm{3}} \:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}+\mathrm{x}^{−\mathrm{2}} }}\:\mathrm{dx}\:= \\ $$$$\int\:\frac{\mathrm{x}^{−\mathrm{3}} −\mathrm{x}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{−\mathrm{2}} +\mathrm{1}}}\:\mathrm{dx}\: \\ $$$$\mathrm{let}\:\mathrm{v}\:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{−\mathrm{2}} +\mathrm{1}\:\rightarrow\mathrm{dv}\:=\:\mathrm{2x}−\mathrm{2x}^{−\mathrm{3}} \:\mathrm{dx} \\ $$$$\mathrm{dv}\:=\:−\mathrm{2}\left(\mathrm{x}^{−\mathrm{3}} −\mathrm{x}\right)\mathrm{dx}\: \\ $$$$ \\ $$$$\int\:\frac{−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{dv}}{\:\sqrt{\mathrm{v}}}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\int\:\mathrm{v}^{−\frac{\mathrm{1}}{\mathrm{2}}} \:\mathrm{dv} \\ $$$$=\:−\sqrt{\mathrm{v}}\:+\:\mathrm{c}\:=\:−\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{−\mathrm{2}} +\mathrm{1}}\:+\:\mathrm{c} \\ $$$$=\:−\frac{\sqrt{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}}}{\mathrm{x}}\:+\:\mathrm{c} \\ $$

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