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Question Number 27805 by abdo imad last updated on 15/Jan/18
find  ∫_1 ^∝   ((arctan(αx))/x^2 ) .
$${find}\:\:\int_{\mathrm{1}} ^{\propto} \:\:\frac{{arctan}\left(\alpha{x}\right)}{{x}^{\mathrm{2}} }\:. \\ $$
Commented by abdo imad last updated on 16/Jan/18
let put I= ∫_1 ^∝  ((arctan(αx))/x^2 )dx  integrate per parts  I=[ −(1/x) arctan(αx)]_1 ^(+∝)  − ∫_1 ^(+∝) −(1/x) (α/(1+α^2 x^2 ))dx  I= artan(α) +α∫_1 ^(+∝)    (dx/(x(1+α^2 x^2 ))) we use the ch. x=(1/α)t  ∫_1 ^(+∝)   (dx/(x(1+α^2 x^2 ))) = ∫_α ^(+∝)   (((1/α)dt)/((1/α)t(1+t^2 ))) = ∫_α ^(+∝)    (dt/(t(1+t^2 )))   but(1/(t (1+t^2 ))) = (1/t) − (t/(1+t^2 )) ⇒ ∫  (dt/(t(1+t^2 )))= ln/t/ −(1/2)ln(1+t^2 ) +k  = ln/(t/( (√(1+t^2 ))))/  so   ∫_α ^(+∝)   (dt/(t(1+t^2 ))) =[ ln/ (t/( (√(1+t^2 ))))/]_α ^(+∝)   =−ln/ (α/( (√(1+α^(2 ) ))))/  I=artan(α)−α ln/ (α/( (√(1+α^2 ))))/  .
$${let}\:{put}\:{I}=\:\int_{\mathrm{1}} ^{\propto} \:\frac{{arctan}\left(\alpha{x}\right)}{{x}^{\mathrm{2}} }{dx}\:\:{integrate}\:{per}\:{parts} \\ $$$${I}=\left[\:−\frac{\mathrm{1}}{{x}}\:{arctan}\left(\alpha{x}\right)\right]_{\mathrm{1}} ^{+\propto} \:−\:\int_{\mathrm{1}} ^{+\propto} −\frac{\mathrm{1}}{{x}}\:\frac{\alpha}{\mathrm{1}+\alpha^{\mathrm{2}} {x}^{\mathrm{2}} }{dx} \\ $$$${I}=\:{artan}\left(\alpha\right)\:+\alpha\int_{\mathrm{1}} ^{+\propto} \:\:\:\frac{{dx}}{{x}\left(\mathrm{1}+\alpha^{\mathrm{2}} {x}^{\mathrm{2}} \right)}\:{we}\:{use}\:{the}\:{ch}.\:{x}=\frac{\mathrm{1}}{\alpha}{t} \\ $$$$\int_{\mathrm{1}} ^{+\propto} \:\:\frac{{dx}}{{x}\left(\mathrm{1}+\alpha^{\mathrm{2}} {x}^{\mathrm{2}} \right)}\:=\:\int_{\alpha} ^{+\propto} \:\:\frac{\frac{\mathrm{1}}{\alpha}{dt}}{\frac{\mathrm{1}}{\alpha}{t}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:=\:\int_{\alpha} ^{+\propto} \:\:\:\frac{{dt}}{{t}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} \\ $$$$\:{but}\frac{\mathrm{1}}{{t}\:\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:=\:\frac{\mathrm{1}}{{t}}\:−\:\frac{{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\Rightarrow\:\int\:\:\frac{{dt}}{{t}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}=\:{ln}/{t}/\:−\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\:+{k} \\ $$$$=\:{ln}/\frac{{t}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}/\:\:{so}\:\:\:\int_{\alpha} ^{+\propto} \:\:\frac{{dt}}{{t}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:=\left[\:{ln}/\:\frac{{t}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}/\right]_{\alpha} ^{+\propto} \\ $$$$=−{ln}/\:\frac{\alpha}{\:\sqrt{\mathrm{1}+\alpha^{\mathrm{2}\:} }}/ \\ $$$${I}={artan}\left(\alpha\right)−\alpha\:{ln}/\:\frac{\alpha}{\:\sqrt{\mathrm{1}+\alpha^{\mathrm{2}} }}/\:\:.\: \\ $$$$ \\ $$

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