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Question Number 137190 by mnjuly1970 last updated on 30/Mar/21

$$....advanced.......calculus.....$$$$provethat:::$$$$\mathit{\varphi }={\int }_0^{1}ln\left(ln\left(\frac{1}{x}\right)\right).\frac{dx}{\sqrt{ln\left(\frac{1}{x}\right)}}=-\sqrt{\pi }(\gamma +ln\left(4\right))$$$$$$

Answered by Dwaipayan Shikari last updated on 30/Mar/21

$$log\left(\frac{1}{x}\right)=u\Rightarrow x=e^{-u}\Rightarrow 1=-e^{-u}\frac{du}{dx}$$$$=-{\int }_{\mathrm{\infty }}^0{t}^{-\frac{1}{2}}{e}^{-u}log\left(t\right)dt$$$$={\mathrm{\Gamma }}^{\prime }\left(\frac{1}{2}\right)=-\sqrt{\pi }(\gamma +log\left(4\right))$$

Commented by mnjuly1970 last updated on 30/Mar/21

$$thankyouverymuch...$$

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