Question Number 66971 by Cmr 237 last updated on 21/Aug/19
$$\int\frac{\mathrm{dx}}{\mathrm{x}\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\:}}=? \\ $$$$\boldsymbol{\mathrm{p}\mathfrak{l}}\mathrm{ease}\:\mathrm{help} \\ $$
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](https://www.tinkutara.com/question/Q66989.png)
$$\int\frac{{dx}}{{x}\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\mathrm{1}}{{x}}\:\rightarrow\:{dx}=−{x}^{\mathrm{2}} {dt}\right] \\ $$$$=−\int\frac{{dt}}{\:\sqrt{{t}^{\mathrm{2}} +{t}+\mathrm{1}}}= \\ $$$$\:\:\:\:\:\left[{u}=\mathrm{2}{t}+\mathrm{1}\:\rightarrow\:{dt}=\frac{{du}}{\mathrm{2}}\right] \\ $$$$=−\int\frac{{du}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{3}}} \\ $$$$\mathrm{now}\:\mathrm{it}\:\mathrm{should}\:\mathrm{be}\:\mathrm{easy} \\ $$
Commented by Cmr 237 last updated on 21/Aug/19
$$\boldsymbol{\mathfrak{t}}\mathrm{hank}\:\mathrm{sir} \\ $$
Commented by mathmax by abdo last updated on 23/Aug/19
$${this}\:{integral}\:{is}\:{solved}\:{see}\:{the}\:{Q}\:\mathrm{66938} \\ $$