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Question Number 131520 by EDWIN88 last updated on 05/Feb/21

 Nice integral   ∫_0 ^( ∞)  (x^2 /((x+100)^3 )) dx =?

$$\:\mathrm{Nice}\:\mathrm{integral}\: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{x}+\mathrm{100}\right)^{\mathrm{3}} }\:\mathrm{dx}\:=? \\ $$

Commented by liberty last updated on 05/Feb/21

diverges

$$\mathrm{diverges} \\ $$

Commented by Dwaipayan Shikari last updated on 05/Feb/21

I think it should be ∫_0 ^∞ (x^2 /((x^2 +100)^3 ))dx  then  (1/2)∫_0 ^∞ (u^(1/2) /((u+100)^3 ))du     u=100t  =((100.10)/(2.100^3 ))∫_0 ^∞ (t^((3/2)−1) /((t+1)^((3/2)+(3/2)) ))dt=(1/(2000)).((Γ((3/2))Γ((3/2)))/(Γ(3)))=(π/(16000))

$${I}\:{think}\:{it}\:{should}\:{be}\:\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} +\mathrm{100}\right)^{\mathrm{3}} }{dx} \\ $$$${then} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \frac{{u}^{\frac{\mathrm{1}}{\mathrm{2}}} }{\left({u}+\mathrm{100}\right)^{\mathrm{3}} }{du}\:\:\:\:\:{u}=\mathrm{100}{t} \\ $$$$=\frac{\mathrm{100}.\mathrm{10}}{\mathrm{2}.\mathrm{100}^{\mathrm{3}} }\int_{\mathrm{0}} ^{\infty} \frac{{t}^{\frac{\mathrm{3}}{\mathrm{2}}−\mathrm{1}} }{\left({t}+\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}+\frac{\mathrm{3}}{\mathrm{2}}} }{dt}=\frac{\mathrm{1}}{\mathrm{2000}}.\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)}{\Gamma\left(\mathrm{3}\right)}=\frac{\pi}{\mathrm{16000}} \\ $$

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