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find-I-n-0-dx-x-2-1-x-2-2-x-2-n-




Question Number 147006 by mathmax by abdo last updated on 17/Jul/21
find  I_n =∫_0 ^∞    (dx/((x^2 +1)(x^2  +2)......(x^2  +n)))
$$\mathrm{find}\:\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{2}\right)……\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{n}\right)} \\ $$
Answered by mindispower last updated on 17/Jul/21
I_n =(1/2)∫_(−∞) ^∞ (dx/(Π_(k=1) ^n (x^2 +(√k^2 ))))  by Residue th in uper half ?complex plan  Im(z)≥0  =2iπΣ_(k=1) ^n Res(f,i(√k))=Σ_(k=1) ^n 2iπ,(1/(2i(√k))).Π_(j=1,j≠k) ^n (1/((k−j)))  =Σ_(k=1) ^n (π/( (√k))),Π_(j=1,j≠k) ^n (1/((j−k)))
$${I}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\int_{−\infty} ^{\infty} \frac{{dx}}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left({x}^{\mathrm{2}} +\sqrt{{k}^{\mathrm{2}} }\right)} \\ $$$${by}\:{Residue}\:{th}\:{in}\:{uper}\:{half}\:?{complex}\:{plan} \\ $$$${Im}\left({z}\right)\geqslant\mathrm{0} \\ $$$$=\mathrm{2}{i}\pi\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{Res}\left({f},{i}\sqrt{{k}}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{2}{i}\pi,\frac{\mathrm{1}}{\mathrm{2}{i}\sqrt{{k}}}.\underset{{j}=\mathrm{1},{j}\neq{k}} {\overset{{n}} {\prod}}\frac{\mathrm{1}}{\left({k}−{j}\right)} \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\pi}{\:\sqrt{{k}}},\underset{{j}=\mathrm{1},{j}\neq{k}} {\overset{{n}} {\prod}}\frac{\mathrm{1}}{\left({j}−{k}\right)} \\ $$

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