Question Number 66048 by aliesam last updated on 08/Aug/19
![∫(x/( (√(ln(1/x))))) dx](https://www.tinkutara.com/question/Q66048.png)
$$\int\frac{{x}}{\:\sqrt{{ln}\left(\mathrm{1}/{x}\right)}}\:{dx} \\ $$
Commented by Prithwish sen last updated on 08/Aug/19
![∫(x/( (√(−lnx))))dx put−lnx = u^2 ⇒dx=−2ue^(−u^2 ) = −2∫e^(−2u^2 ) du and it is a Gaussian integral. And it has no closed form. please check.](https://www.tinkutara.com/question/Q66050.png)
$$\int\frac{\mathrm{x}}{\:\sqrt{−\mathrm{lnx}}}\mathrm{dx}\:\:\:\mathrm{put}−\mathrm{lnx}\:=\:\mathrm{u}^{\mathrm{2}} \:\Rightarrow\mathrm{dx}=−\mathrm{2ue}^{−\mathrm{u}^{\mathrm{2}} } \\ $$$$=\:−\mathrm{2}\int\mathrm{e}^{−\mathrm{2u}^{\mathrm{2}} } \mathrm{du}\:\:\:\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{Gaussian}\:\mathrm{integral}. \\ $$$$\mathrm{And}\:\mathrm{it}\:\mathrm{has}\:\mathrm{no}\:\mathrm{closed}\:\mathrm{form}.\: \\ $$$$\mathrm{please}\:\mathrm{check}. \\ $$