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Question Number 30215 by abdo imad last updated on 18/Feb/18

let give J(x)= (1/π) ∫_0 ^π cos(xcost)dt  1) find J^′  and J^(′′)  in form of integrals  2)prove that J^′ (x)=((−x)/π) ∫_0 ^π  sin^2 t cos(xcost)dt and J is  solution of d.e.  xy^(′′)  +y^′  +xy=0

$${let}\:{give}\:{J}\left({x}\right)=\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {cos}\left({xcost}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{J}^{'} \:{and}\:{J}^{''} \:{in}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{J}^{'} \left({x}\right)=\frac{−{x}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{2}} {t}\:{cos}\left({xcost}\right){dt}\:{and}\:{J}\:{is} \\ $$$${solution}\:{of}\:{d}.{e}.\:\:{xy}^{''} \:+{y}^{'} \:+{xy}=\mathrm{0} \\ $$

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