Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 28610 by abdo imad last updated on 27/Jan/18

let give I(x)= ∫_0 ^(π/2)    (dt/(√(sin^2 t +x^2  cos^2 t)))  and  J(x)= ∫_0 ^(π/2)   ((cost)/(√(sin^2 t +x^2 cos^2 t)))dt cslculate lim_(x→0^+ ) (I(x)−J(x))  and prove that  I(x)=_(x→0^+ )  −lnx +2ln2 +o(1).

$${let}\:{give}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} \:{cos}^{\mathrm{2}} {t}}}\:\:{and} \\ $$$${J}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cost}}{\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} {cos}^{\mathrm{2}} {t}}}{dt}\:{cslculate}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \left({I}\left({x}\right)−{J}\left({x}\right)\right) \\ $$$${and}\:{prove}\:{that}\:\:{I}\left({x}\right)=_{{x}\rightarrow\mathrm{0}^{+} } \:−{lnx}\:+\mathrm{2}{ln}\mathrm{2}\:+{o}\left(\mathrm{1}\right). \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com