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Question Number 96554 by student work last updated on 02/Jun/20

∫(dx/(x!))=?

$$\int\frac{\mathrm{dx}}{\mathrm{x}!}=? \\ $$

Commented by student work last updated on 02/Jun/20

or ∫(x/(x!))dx=?

$$\mathrm{or}\:\int\frac{\mathrm{x}}{\mathrm{x}!}\mathrm{dx}=? \\ $$

Commented by MJS last updated on 03/Jun/20

I stated this before x! is defined for x∈N if you mean Γ (x) you must say Γ (x)

$$\mathrm{I}\:\mathrm{stated}\:\mathrm{this}\:\mathrm{before} \\ $$$${x}!\:\mathrm{is}\:\mathrm{defined}\:\mathrm{for}\:{x}\in\mathbb{N} \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{mean}\:\Gamma\:\left({x}\right)\:\mathrm{you}\:\mathrm{must}\:\mathrm{say}\:\Gamma\:\left({x}\right) \\ $$

Answered by Rio Michael last updated on 03/Jun/20

recall x! = Γ(x +1) ⇒ ∫(dx/(x!)) = ∫(1/(Γ(x + 1))) dx = x + ((γx^2 )/2) + (1/(36))(6γ^2 −π^2 )x^3 + (1/(48))x^4 (2γ^3 −γπ^2 −2ψ^((2)) (1)) + (1/(7200)) x^5 (60γ^4 + + π^4 −60γ^2 π^2 −240ψγ^((2)) (1))+ (1/(8467420))x^7 (168γ^4 + 42γ^2 π^4 −420γ^4 π^4 −3360γ^3 ψ^((2)) (1)) −5(π^6 −336ψ^((2)) (1)^((2)) ) + 336γ(5π^2 ψ^((2)) (1) −3ψ^((4)) (1)) + (1/(8640))x^6 (12γ^5 + γπ^4 −20γ^3 π^2 −120γ^2 ψ^((2)) (1) + 20π^2 ψ^((2)) (1) −12ψ^((4)) (1)) +...k γ = Euler−Mascheroni constant ψ^((m)) (z) = polygama function summarising the above, we see that ∫(1/(Γ(x + 1)))dx = (i/(2π))∫(∮_H (−t)^(−(1 + x)) e^(−t) dt)dx. also please verify on wolframe alpha that (1/(Γ(x + 1))) = Σ_(k = 1) ^∞ c_k (x + 1)^k , c_1 = 1 , c_2 = 2 and c_k = ((γc_(k−1) + Σ_(j=2) ^(k−2) c_j (−1)^(j + k +1) ζ(k−j))/(k−1)) ζ(z) = Reiman zeta function integrating the series above ∫(1/(Γ(1 +x))) dx = ∫(Σ_(k=1) ^∞ c_k (x + 1)^k )dx = Σ_(k=1) ^∞ c_k ∫(x + 1)^k dx ∫(1/(Γ(1 + x)))dx = ∫(dx/(x!)) = Σ_(k=1) ^∞ ((c_k (x + 1)^(k+1) )/(k + 1)) + constant (A)

$$\mathrm{recall}\:{x}!\:=\:\Gamma\left({x}\:+\mathrm{1}\right) \\ $$$$\Rightarrow\:\int\frac{{dx}}{{x}!}\:=\:\int\frac{\mathrm{1}}{\Gamma\left({x}\:+\:\mathrm{1}\right)}\:{dx}\:= \\ $$$${x}\:+\:\frac{\gamma{x}^{\mathrm{2}} }{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{36}}\left(\mathrm{6}\gamma^{\mathrm{2}} −\pi^{\mathrm{2}} \right){x}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\mathrm{48}}{x}^{\mathrm{4}} \left(\mathrm{2}\gamma^{\mathrm{3}} −\gamma\pi^{\mathrm{2}} −\mathrm{2}\psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\right)\:+\:\frac{\mathrm{1}}{\mathrm{7200}}\:{x}^{\mathrm{5}} \left(\mathrm{60}\gamma^{\mathrm{4}} \:+\right. \\ $$$$\left.+\:\pi^{\mathrm{4}} \:−\mathrm{60}\gamma^{\mathrm{2}} \pi^{\mathrm{2}} −\mathrm{240}\psi\gamma^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\right)+\:\frac{\mathrm{1}}{\mathrm{8467420}}{x}^{\mathrm{7}} \left(\mathrm{168}\gamma^{\mathrm{4}} \:+\:\mathrm{42}\gamma^{\mathrm{2}} \pi^{\mathrm{4}} −\mathrm{420}\gamma^{\mathrm{4}} \pi^{\mathrm{4}} −\mathrm{3360}\gamma^{\mathrm{3}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\right) \\ $$$$\:−\mathrm{5}\left(\pi^{\mathrm{6}} −\mathrm{336}\psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)^{\left(\mathrm{2}\right)} \right)\:+\:\mathrm{336}\gamma\left(\mathrm{5}\pi^{\mathrm{2}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\:−\mathrm{3}\psi^{\left(\mathrm{4}\right)} \left(\mathrm{1}\right)\right)\:+\:\frac{\mathrm{1}}{\mathrm{8640}}{x}^{\mathrm{6}} \left(\mathrm{12}\gamma^{\mathrm{5}} \:+\right. \\ $$$$\left.\:\:\gamma\pi^{\mathrm{4}} −\mathrm{20}\gamma^{\mathrm{3}} \pi^{\mathrm{2}} −\mathrm{120}\gamma^{\mathrm{2}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\:+\:\mathrm{20}\pi^{\mathrm{2}} \psi^{\left(\mathrm{2}\right)} \left(\mathrm{1}\right)\:−\mathrm{12}\psi^{\left(\mathrm{4}\right)} \left(\mathrm{1}\right)\right)\:+...{k}\:\: \\ $$$$\gamma\:=\:\mathrm{Euler}−\mathrm{Mascheroni}\:\mathrm{constant} \\ $$$$\psi^{\left({m}\right)} \left({z}\right)\:=\:\mathrm{polygama}\:\mathrm{function} \\ $$$$\mathrm{summarising}\:\mathrm{the}\:\mathrm{above},\:\mathrm{we}\:\mathrm{see}\:\mathrm{that}\: \\ $$$$\:\int\frac{\mathrm{1}}{\Gamma\left({x}\:+\:\mathrm{1}\right)}{dx}\:=\:\frac{{i}}{\mathrm{2}\pi}\int\left(\oint_{{H}} \left(−{t}\right)^{−\left(\mathrm{1}\:+\:{x}\right)} {e}^{−{t}} \:{dt}\right){dx}. \\ $$$$\mathrm{also}\:\mathrm{please}\:\mathrm{verify}\:\mathrm{on}\:\mathrm{wolframe}\:\mathrm{alpha}\:\mathrm{that}\: \\ $$$$\:\frac{\mathrm{1}}{\Gamma\left({x}\:+\:\mathrm{1}\right)}\:=\:\underset{{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{k}} \left({x}\:+\:\mathrm{1}\right)^{{k}} ,\: \\ $$$${c}_{\mathrm{1}} \:=\:\mathrm{1}\:,\:{c}_{\mathrm{2}} \:=\:\mathrm{2}\:\mathrm{and}\:{c}_{{k}} \:=\:\frac{\gamma{c}_{{k}−\mathrm{1}} \:+\:\underset{{j}=\mathrm{2}} {\overset{{k}−\mathrm{2}} {\sum}}{c}_{{j}} \left(−\mathrm{1}\right)^{{j}\:+\:{k}\:+\mathrm{1}} \zeta\left({k}−{j}\right)}{{k}−\mathrm{1}} \\ $$$$\zeta\left({z}\right)\:=\:\mathrm{Reiman}\:\mathrm{zeta}\:\mathrm{function} \\ $$$$\mathrm{integrating}\:\mathrm{the}\:\mathrm{series}\:\mathrm{above}\: \\ $$$$\int\frac{\mathrm{1}}{\Gamma\left(\mathrm{1}\:+{x}\right)}\:{dx}\:=\:\int\left(\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{k}} \left({x}\:+\:\mathrm{1}\right)^{{k}} \right){dx}\:=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}{c}_{{k}} \int\left({x}\:+\:\mathrm{1}\right)^{{k}} {dx}\: \\ $$$$\:\int\frac{\mathrm{1}}{\Gamma\left(\mathrm{1}\:+\:{x}\right)}{dx}\:=\:\int\frac{{dx}}{{x}!}\:=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{c}_{{k}} \left({x}\:+\:\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}\:+\:\mathrm{1}}\:+\:\mathrm{constant}\:\left({A}\right) \\ $$$$ \\ $$

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