Question Number 96554 by student work last updated on 02/Jun/20
$$\int\frac{\mathrm{dx}}{\mathrm{x}!}=? \\ $$
Commented by student work last updated on 02/Jun/20
$$\mathrm{or}\:\int\frac{\mathrm{x}}{\mathrm{x}!}\mathrm{dx}=? \\ $$
Commented by MJS last updated on 03/Jun/20
$$\mathrm{I}\:\mathrm{stated}\:\mathrm{this}\:\mathrm{before} \\ $$$${x}!\:\mathrm{is}\:\mathrm{defined}\:\mathrm{for}\:{x}\in\mathbb{N} \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{mean}\:\Gamma\:\left({x}\right)\:\mathrm{you}\:\mathrm{must}\:\mathrm{say}\:\Gamma\:\left({x}\right) \\ $$
Answered by Rio Michael last updated on 03/Jun/20
$$\mathrm{recall}\:{x}!\:=\:\Gamma\left({x}\:+\mathrm{1}\right) \\ $$$$\Rightarrow\:\int\frac{{dx}}{{x}!}\:=\:\int\frac{\mathrm{1}}{\Gamma\left({x}\:+\:\mathrm{1}\right)}\:{dx}\:= \\ $$$${x}\:+\:\frac{\gamma{x}^{\mathrm{2}} }{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{36}}\left(\mathrm{6}\gamma^{\mathrm{2}} −\pi^{\mathrm{2}} \right){x}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\mathrm{48}}{x}^{\mathrm{4}} \left(\mathrm{2}\gamma^{\mathrm{3}} −\gamma\pi^{\mathrm{2}} −\mathrm{2}\psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\right)\:+\:\frac{\mathrm{1}}{\mathrm{7200}}\:{x}^{\mathrm{5}} \left(\mathrm{60}\gamma^{\mathrm{4}} \:+\right. \\ $$$$\left.+\:\pi^{\mathrm{4}} \:−\mathrm{60}\gamma^{\mathrm{2}} \pi^{\mathrm{2}} −\mathrm{240}\psi\gamma^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\right)+\:\frac{\mathrm{1}}{\mathrm{8467420}}{x}^{\mathrm{7}} \left(\mathrm{168}\gamma^{\mathrm{4}} \:+\:\mathrm{42}\gamma^{\mathrm{2}} \pi^{\mathrm{4}} −\mathrm{420}\gamma^{\mathrm{4}} \pi^{\mathrm{4}} −\mathrm{3360}\gamma^{\mathrm{3}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\right) \\ $$$$\:−\mathrm{5}\left(\pi^{\mathrm{6}} −\mathrm{336}\psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)^{\left(\mathrm{2}\right)} \right)\:+\:\mathrm{336}\gamma\left(\mathrm{5}\pi^{\mathrm{2}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\:−\mathrm{3}\psi^{\left(\mathrm{4}\right)} \left(\mathrm{1}\right)\right)\:+\:\frac{\mathrm{1}}{\mathrm{8640}}{x}^{\mathrm{6}} \left(\mathrm{12}\gamma^{\mathrm{5}} \:+\right. \\ $$$$\left.\:\:\gamma\pi^{\mathrm{4}} −\mathrm{20}\gamma^{\mathrm{3}} \pi^{\mathrm{2}} −\mathrm{120}\gamma^{\mathrm{2}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\:+\:\mathrm{20}\pi^{\mathrm{2}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\:−\mathrm{12}\psi^{\left(\mathrm{4}\right)} \left(\mathrm{1}\right)\right)\:+…{k}\:\: \\ $$$$\gamma\:=\:\mathrm{Euler}−\mathrm{Mascheroni}\:\mathrm{constant} \\ $$$$\psi^{\left({m}\right)} \left({z}\right)\:=\:\mathrm{polygama}\:\mathrm{function} \\ $$$$\mathrm{summarising}\:\mathrm{the}\:\mathrm{above},\:\mathrm{we}\:\mathrm{see}\:\mathrm{that}\: \\ $$$$\:\int\frac{\mathrm{1}}{\Gamma\left({x}\:+\:\mathrm{1}\right)}{dx}\:=\:\frac{{i}}{\mathrm{2}\pi}\int\left(\oint_{{H}} \left(−{t}\right)^{−\left(\mathrm{1}\:+\:{x}\right)} {e}^{−{t}} \:{dt}\right){dx}. \\ $$$$\mathrm{also}\:\mathrm{please}\:\mathrm{verify}\:\mathrm{on}\:\mathrm{wolframe}\:\mathrm{alpha}\:\mathrm{that}\: \\ $$$$\:\frac{\mathrm{1}}{\Gamma\left({x}\:+\:\mathrm{1}\right)}\:=\:\underset{{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{k}} \left({x}\:+\:\mathrm{1}\right)^{{k}} ,\: \\ $$$${c}_{\mathrm{1}} \:=\:\mathrm{1}\:,\:{c}_{\mathrm{2}} \:=\:\mathrm{2}\:\mathrm{and}\:{c}_{{k}} \:=\:\frac{\gamma{c}_{{k}−\mathrm{1}} \:+\:\underset{{j}=\mathrm{2}} {\overset{{k}−\mathrm{2}} {\sum}}{c}_{{j}} \left(−\mathrm{1}\right)^{{j}\:+\:{k}\:+\mathrm{1}} \zeta\left({k}−{j}\right)}{{k}−\mathrm{1}} \\ $$$$\zeta\left({z}\right)\:=\:\mathrm{Reiman}\:\mathrm{zeta}\:\mathrm{function} \\ $$$$\mathrm{integrating}\:\mathrm{the}\:\mathrm{series}\:\mathrm{above}\: \\ $$$$\int\frac{\mathrm{1}}{\Gamma\left(\mathrm{1}\:+{x}\right)}\:{dx}\:=\:\int\left(\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{k}} \left({x}\:+\:\mathrm{1}\right)^{{k}} \right){dx}\:=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{k}} \int\left({x}\:+\:\mathrm{1}\right)^{{k}} {dx}\: \\ $$$$\:\int\frac{\mathrm{1}}{\Gamma\left(\mathrm{1}\:+\:{x}\right)}{dx}\:=\:\int\frac{{dx}}{{x}!}\:=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{c}_{{k}} \left({x}\:+\:\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}\:+\:\mathrm{1}}\:+\:\mathrm{constant}\:\left({A}\right) \\ $$$$ \\ $$