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Question Number 67974 by mathmax by abdo last updated on 02/Sep/19
let F(x) =∫_x ^x^2 ((arctan(xt))/(x^2 +t^2 ))dt calculate F^′ (x).
$${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{arctan}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }{dt}\:\:{calculate}\:{F}\:^{'} \left({x}\right). \\ $$
Commented by mathmax by abdo last updated on 03/Sep/19
we have g(x,t) =((arctan(xt))/(x^2 +t^2 )) ,u(x) =x and v(x)=x^2 we use the formulae F^′ (x) =∫_(u(x)) ^(v(x)) (∂g/∂x)(x,t)dt +v^′ g(x,v) −u^′ g(x,u) =∫_x ^x^2 (∂g/∂x)(x,t)dt +2x ×((arctan(x^3 ))/(x^2 +x^4 )) −((arctan(x^2 ))/(2x^2 )) (∂g/∂x)(x,t) = (((t/(1+x^2 t^2 ))(x^2 +t^2 )−2x arctan(xt))/((x^2 +t^2 )^2 ))
$${we}\:{have}\:{g}\left({x},{t}\right)\:=\frac{{arctan}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }\:,{u}\left({x}\right)\:={x}\:\:{and}\:{v}\left({x}\right)={x}^{\mathrm{2}} \: \\ $$$${we}\:{use}\:{the}\:{formulae}\:\:{F}\:^{'} \left({x}\right)\:=\int_{{u}\left({x}\right)} ^{{v}\left({x}\right)} \:\frac{\partial{g}}{\partial{x}}\left({x},{t}\right){dt}\:+{v}^{'} {g}\left({x},{v}\right) \\ $$$$−{u}^{'} {g}\left({x},{u}\right)\:\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \frac{\partial{g}}{\partial{x}}\left({x},{t}\right){dt}\:+\mathrm{2}{x}\:×\frac{{arctan}\left({x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} } \\ $$$$−\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{\mathrm{2}{x}^{\mathrm{2}} } \\ $$$$\frac{\partial{g}}{\partial{x}}\left({x},{t}\right)\:=\:\frac{\frac{{t}}{\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} }\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)−\mathrm{2}{x}\:{arctan}\left({xt}\right)}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$ \\ $$
Commented by mathmax by abdo last updated on 03/Sep/19
⇒ F^′ (x) =∫_x ^x^2 ( (t/((1+x^2 t^2 )(x^(2 ) +t^2 ))) −((2x arctan(xt))/((x^2 +t^2 )^2 )))dt +((2xarctan(x^3 ))/(x^2 +x^4 )) −((arctan(x^2 ))/(2x^2 )) ....be continued...
$$\Rightarrow\:{F}\:^{'} \left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \left(\:\:\frac{{t}}{\left(\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}\:} +{t}^{\mathrm{2}} \right)}\:−\frac{\mathrm{2}{x}\:{arctan}\left({xt}\right)}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\right){dt} \\ $$$$+\frac{\mathrm{2}{xarctan}\left({x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} }\:−\frac{{arctan}\left({x}^{\mathrm{2}} \right)}{\mathrm{2}{x}^{\mathrm{2}} }\:….{be}\:{continued}… \\ $$

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