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Question Number 54821 by Abdo msup. last updated on 12/Feb/19
find lim_(n→+∞)    ∫_0 ^n    ((arctan(nx))/(n^2  +x^2 ))dx
$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\int_{\mathrm{0}} ^{{n}} \:\:\:\frac{{arctan}\left({nx}\right)}{{n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }{dx} \\ $$
Commented by maxmathsup by imad last updated on 13/Feb/19
let A_n =∫_0 ^n   ((arctan(nx))/(n^2  +x^2 ))dx ⇒A_n =_(x=nt)       ∫_0 ^1   ((arctan(n^2 t))/(n^2 (1+t^2 ))) ndt  =(1/n) ∫_0 ^1    ((arctan(n^2 t))/(1+t^2 )) dt  but  lim_(n→+∞)     ∫_0 ^1   ((arctan(n^2 t))/(1+t^2 )) dt =(π/2) ∫_0 ^1  (dt/(1+t^2 ))  =(π/2) (π/4) =(π^2 /8)  and lim_(n→+∞)  (1/n) =0 ⇒ lim_(n→+∞)  A_n =0 .
$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{{n}} \:\:\frac{{arctan}\left({nx}\right)}{{n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }{dx}\:\Rightarrow{A}_{{n}} =_{{x}={nt}} \:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({n}^{\mathrm{2}} {t}\right)}{{n}^{\mathrm{2}} \left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:{ndt} \\ $$$$=\frac{\mathrm{1}}{{n}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left({n}^{\mathrm{2}} {t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:\:{but}\:\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({n}^{\mathrm{2}} {t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:=\frac{\pi}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$=\frac{\pi}{\mathrm{2}}\:\frac{\pi}{\mathrm{4}}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}}\:=\mathrm{0}\:\Rightarrow\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\mathrm{0}\:. \\ $$

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