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Question Number 55615 by Abdo msup. last updated on 28/Feb/19

let F(α)=∫_α ^(1+α^2 ) ((sin(αx))/(1+αx^2 ))dx 1) calculate (dF/dα)(α) 2) calculate lim_(α→0) F(α)

$${let}\:{F}\left(\alpha\right)=\int_{\alpha} ^{\mathrm{1}+\alpha^{\mathrm{2}} } \:\:\frac{{sin}\left(\alpha{x}\right)}{\mathrm{1}+\alpha{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\frac{{dF}}{{d}\alpha}\left(\alpha\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:{lim}_{\alpha\rightarrow\mathrm{0}} \:\:{F}\left(\alpha\right) \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Feb/19

(dF/dα)=∫_α ^(1+α^2 ) (∂/∂α)(((sinαx)/(1+αx^2 )))dx +((sinα(1+α^2 ))/(1+α(1+α)^2 ))(d/dα)(1+α^2 )−((sinα(α))/(1+α(α)^2 ))×(d/dα)(α) =∫_α ^(1+α^2 ) (((1+αx^2 )×cosαx×x−sinαx(0+x^2 ))/((1+αx^2 )^2 ))dx+((sin(α+α^3 ))/(1+α+2α^2 +α^3 ))×(2α)−((sinα^2 )/(1+α^3 ))×1 =∫_α ^(1+α^2 ) ((xcosαx+αx^3 cosαx^2 −x^2 sinαx)/((1+αx^2 )^2 ))dx+((2αsin(α+α^3 ))/(1+α+2α^2 +α^3 ))−((sinα^2 )/(1+α^3 )) wait...

$$\frac{{dF}}{{d}\alpha}=\int_{\alpha} ^{\mathrm{1}+\alpha^{\mathrm{2}} } \frac{\partial}{\partial\alpha}\left(\frac{{sin}\alpha{x}}{\mathrm{1}+\alpha{x}^{\mathrm{2}} }\right){dx}\:+\frac{{sin}\alpha\left(\mathrm{1}+\alpha^{\mathrm{2}} \right)}{\mathrm{1}+\alpha\left(\mathrm{1}+\alpha\right)^{\mathrm{2}} }\frac{{d}}{{d}\alpha}\left(\mathrm{1}+\alpha^{\mathrm{2}} \right)−\frac{{sin}\alpha\left(\alpha\right)}{\mathrm{1}+\alpha\left(\alpha\right)^{\mathrm{2}} }×\frac{{d}}{{d}\alpha}\left(\alpha\right) \\ $$$$=\int_{\alpha} ^{\mathrm{1}+\alpha^{\mathrm{2}} } \frac{\left(\mathrm{1}+\alpha{x}^{\mathrm{2}} \right)×{cos}\alpha{x}×{x}−{sin}\alpha{x}\left(\mathrm{0}+{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+\alpha{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}+\frac{{sin}\left(\alpha+\alpha^{\mathrm{3}} \right)}{\mathrm{1}+\alpha+\mathrm{2}\alpha^{\mathrm{2}} +\alpha^{\mathrm{3}} }×\left(\mathrm{2}\alpha\right)−\frac{{sin}\alpha^{\mathrm{2}} }{\mathrm{1}+\alpha^{\mathrm{3}} }×\mathrm{1} \\ $$$$=\int_{\alpha} ^{\mathrm{1}+\alpha^{\mathrm{2}} } \frac{{xcos}\alpha{x}+\alpha{x}^{\mathrm{3}} {cos}\alpha{x}^{\mathrm{2}} \:−{x}^{\mathrm{2}} {sin}\alpha{x}}{\left(\mathrm{1}+\alpha{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}+\frac{\mathrm{2}\alpha{sin}\left(\alpha+\alpha^{\mathrm{3}} \right)}{\mathrm{1}+\alpha+\mathrm{2}\alpha^{\mathrm{2}} +\alpha^{\mathrm{3}} }−\frac{{sin}\alpha^{\mathrm{2}} }{\mathrm{1}+\alpha^{\mathrm{3}} } \\ $$$${wait}... \\ $$

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