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Question Number 37600 by prof Abdo imad last updated on 15/Jun/18

n integr natural calculate ∫_0 ^∞ (dx/((x+1)(x+2)......(x+n)))

$${n}\:{integr}\:{natural}\:{calculate} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)......\left({x}+{n}\right)} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 16/Jun/18

(1/((x+1)(x+2)(x+3)...(x+n)))=e^t −{ln(x+1)+ln(x+2)+ln(x+3)..+ln(x+n)= t dt=−{(1/(x+1))+(1/(x+2))+...+(1/(x+n))}dx =−{(((e^t /(x+1))+(e^t /(x+2))+...+(e^t /(x+n)))/((x+1)(x+2)(x+3)..(x+n)))}contd

$$\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)...\left({x}+{n}\right)}={e}^{{t}} \\ $$$$−\left\{{ln}\left({x}+\mathrm{1}\right)+{ln}\left({x}+\mathrm{2}\right)+{ln}\left({x}+\mathrm{3}\right)..+{ln}\left({x}+{n}\right)=\right. \\ $$$$\:\:\:{t} \\ $$$${dt}=−\left\{\frac{\mathrm{1}}{{x}+\mathrm{1}}+\frac{\mathrm{1}}{{x}+\mathrm{2}}+...+\frac{\mathrm{1}}{{x}+{n}}\right\}{dx} \\ $$$$=−\left\{\frac{\frac{{e}^{{t}} }{{x}+\mathrm{1}}+\frac{{e}^{{t}} }{{x}+\mathrm{2}}+...+\frac{{e}^{{t}} }{{x}+{n}}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)..\left({x}+{n}\right)}\right\}{contd} \\ $$

Commented by math khazana by abdo last updated on 17/Jun/18

you are really a rail ways...

$${you}\:{are}\:{really}\:{a}\:{rail}\:{ways}... \\ $$

Answered by ajfour last updated on 17/Jun/18

I=(1/((n−1)!))∫_0 ^( ∞) (dx/(x+1))−(1/((n−2)!))∫_0 ^( ∞) (dx/(x+2)) −(1/(2!(n−3)!))∫_0 ^( ∞) (dx/(x+3))+(1/(3!(n−4)!))∫_0 ^( ∞) (dx/(x+4)) .....−(((−1)^n )/((n−1)!))∫_0 ^( ∞) (dx/(x+n)) =lim_(x→∞) Σ(((−1)^r )/((r−1)!(n−r)!))ln (1+(x/r)) =(1/((n−1)!))Σ(−1)^r lim^(n−1) _(x→∞) C_(r−1) ln (1+(x/r)) .....

$${I}=\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}\int_{\mathrm{0}} ^{\:\:\infty} \frac{{dx}}{{x}+\mathrm{1}}−\frac{\mathrm{1}}{\left({n}−\mathrm{2}\right)!}\int_{\mathrm{0}} ^{\:\:\infty} \frac{{dx}}{{x}+\mathrm{2}} \\ $$$$−\frac{\mathrm{1}}{\mathrm{2}!\left({n}−\mathrm{3}\right)!}\int_{\mathrm{0}} ^{\:\:\infty} \frac{{dx}}{{x}+\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}!\left({n}−\mathrm{4}\right)!}\int_{\mathrm{0}} ^{\:\:\infty} \frac{{dx}}{{x}+\mathrm{4}} \\ $$$$.....−\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}−\mathrm{1}\right)!}\int_{\mathrm{0}} ^{\:\:\infty} \frac{{dx}}{{x}+{n}} \\ $$$$\:\:\:=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\Sigma\frac{\left(−\mathrm{1}\right)^{{r}} }{\left({r}−\mathrm{1}\right)!\left({n}−{r}\right)!}\mathrm{ln}\:\left(\mathrm{1}+\frac{{x}}{{r}}\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}\Sigma\left(−\mathrm{1}\right)^{{r}} \underset{{x}\rightarrow\infty} {\mathrm{lim}^{{n}−\mathrm{1}} }{C}_{{r}−\mathrm{1}} \mathrm{ln}\:\left(\mathrm{1}+\frac{{x}}{{r}}\right) \\ $$$$..... \\ $$

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