Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

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 .

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{{n}} \:\:\frac{{arctan}\left({nx}\right)}{{n}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }{dx}\:\Rightarrow{A}_{{n}} =_{{x}={nt}} \:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({n}^{\mathrm{2}} {t}\right)}{{n}^{\mathrm{2}} \left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:{ndt} \\ $$$$=\frac{\mathrm{1}}{{n}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left({n}^{\mathrm{2}} {t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:\:{but}\:\:{lim}_{{n}\rightarrow+\infty} \:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({n}^{\mathrm{2}} {t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:=\frac{\pi}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\ $$$$=\frac{\pi}{\mathrm{2}}\:\frac{\pi}{\mathrm{4}}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}}\:=\mathrm{0}\:\Rightarrow\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} =\mathrm{0}\:. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com