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Question Number 121649 by mnjuly1970 last updated on 10/Nov/20

$$pleaseevaluate::$$$$i:{\mathrm{\Omega }}_1\stackrel{?}{=}{\int }_0^{\mathrm{\infty }}{e}^{-x}ln\left(\mathrm{\Gamma }(1-e^{-x})\right)dx$$$$ii:{\mathrm{\Omega }}_2\stackrel{?}{=}{\int }_0^{\mathrm{\infty }}{\left(\frac{sin\left(x\right)}{x}\right)}^{3}dx$$$$iii::\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n}{n^2}cos\left(\left(\sqrt{n^2{\pi }^2-9}\right)\right)$$$$..................$$

Commented by Dwaipayan Shikari last updated on 10/Nov/20

$${\int }_0^{\mathrm{\infty }}{e}^{-x}\mathrm{\Gamma }(1-e^{-x})dx$$$$={\int }_0^1\mathrm{\Gamma }\left(t\right)dt(1-e^{-x})=t$$$$Itseemstobedivergent..$$

Commented by mnjuly1970 last updated on 10/Nov/20

$$thankyoumrDwaypayan$$$$icorrectedmymistake.$$$${\int }_0^{\mathrm{\infty }}{e}^{-x}ln\left(\mathrm{\Gamma }(1-e^{-x})\right)dx...$$

Answered by mnjuly1970 last updated on 11/Nov/20

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