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Question Number 66971 by Cmr 237 last updated on 21/Aug/19

∫(dx/(x(√(x^2 +x+1 ))))=?  please help

$$\int\frac{\mathrm{dx}}{\mathrm{x}\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\:}}=? \\ $$$$\boldsymbol{\mathrm{p}\mathfrak{l}}\mathrm{ease}\:\mathrm{help} \\ $$

Commented by MJS last updated on 21/Aug/19

∫(dx/(x(√(x^2 +x+1))))=       [t=(1/x) → dx=−x^2 dt]  =−∫(dt/(√(t^2 +t+1)))=       [u=2t+1 → dt=(du/2)]  =−∫(du/(√(u^2 +3)))  now it should be easy

$$\int\frac{{dx}}{{x}\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\mathrm{1}}{{x}}\:\rightarrow\:{dx}=−{x}^{\mathrm{2}} {dt}\right] \\ $$$$=−\int\frac{{dt}}{\sqrt{{t}^{\mathrm{2}} +{t}+\mathrm{1}}}= \\ $$$$\:\:\:\:\:\left[{u}=\mathrm{2}{t}+\mathrm{1}\:\rightarrow\:{dt}=\frac{{du}}{\mathrm{2}}\right] \\ $$$$=−\int\frac{{du}}{\sqrt{{u}^{\mathrm{2}} +\mathrm{3}}} \\ $$$$\mathrm{now}\:\mathrm{it}\:\mathrm{should}\:\mathrm{be}\:\mathrm{easy} \\ $$

Commented by Cmr 237 last updated on 21/Aug/19

thank sir

$$\boldsymbol{\mathfrak{t}}\mathrm{hank}\:\mathrm{sir} \\ $$

Commented by mathmax by abdo last updated on 23/Aug/19

this integral is solved see the Q 66938

$${this}\:{integral}\:{is}\:{solved}\:{see}\:{the}\:{Q}\:\mathrm{66938} \\ $$

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