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Question Number 141062 by mnjuly1970 last updated on 15/May/21

prove that:: φ:=∫_0 ^( ∞) ((e^(cos(x)) sin(sin(x)))/x) dx=(π/2)(e−1)

$$\:\:\:\:\: \\ $$$$\:\:\:\:\:{prove}\:{that}:: \\ $$$$\:\phi:=\int_{\mathrm{0}} ^{\:\infty} \frac{{e}^{{cos}\left({x}\right)} {sin}\left({sin}\left({x}\right)\right)}{{x}}\:{dx}=\frac{\pi}{\mathrm{2}}\left({e}−\mathrm{1}\right) \\ $$

Commented by mindispower last updated on 15/May/21

always pleasur have nice day“ Ai^(..) d Mobarak”

$${always}\:{pleasur}\:{have}\:{nice}\:{day}``\:{A}\overset{..} {{i}d}\:{Mobarak}'' \\ $$

Commented by mnjuly1970 last updated on 15/May/21

thank you so much sir power your Aid mobarak ..

$$\:\:{thank}\:{you}\:{so}\:{much}\:{sir} \\ $$$${power}\:{your}\:{Aid}\:{mobarak}\:.. \\ $$

Commented by mindispower last updated on 15/May/21

starte withe your previous result quation Σ_(n≥1) a^n ((sin(nx))/(n!))=e^(acos(x)) sin(asin(x) φ=∫_0 ^∞ (1/x)Σ_(n≥1) (a^n sin(nx))/(n!))_(a=1) dx =Σ_(n≥1) ∫_0 ^∞ ((sin(nx))/(x(n!)))dx=Σ_(n≥1) (1/(n!))∫_0 ^∞ ((sin(nx))/(nx))d(nx) =Σ_(n≥1) (1/(n!)).(π/2)=(π/2)(Σ_(n≥0) (1/(n!))−1)=(π/2)(e−1)

$${starte}\:{withe}\:{your}\:{previous}\:{result}\:{quation} \\ $$$$\underset{{n}\geqslant\mathrm{1}} {\sum}{a}^{{n}} \frac{{sin}\left({nx}\right)}{{n}!}={e}^{{acos}\left({x}\right)} {sin}\left({asin}\left({x}\right)\right. \\ $$$$\left.\phi=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{{x}}\underset{{n}\geqslant\mathrm{1}} {\sum}\left({a}^{{n}} {sin}\left({nx}\right)\right)/\left({n}!\right)\right)_{{a}=\mathrm{1}} {dx} \\ $$$$=\underset{{n}\geqslant\mathrm{1}} {\sum}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({nx}\right)}{{x}\left({n}!\right)}{dx}=\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}!}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({nx}\right)}{{nx}}{d}\left({nx}\right) \\ $$$$=\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}!}.\frac{\pi}{\mathrm{2}}=\frac{\pi}{\mathrm{2}}\left(\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{{n}!}−\mathrm{1}\right)=\frac{\pi}{\mathrm{2}}\left({e}−\mathrm{1}\right) \\ $$

Commented by mnjuly1970 last updated on 15/May/21

grateful mr power....

$$\:{grateful}\:{mr}\:{power}.... \\ $$

Answered by Dwaipayan Shikari last updated on 15/May/21

e^(cos(x)) sin(sinx)=Σ((sin(nx))/(n!)) ∫_0 ^∞ Σ_(n=1) ((sin(nx))/(xn!))=(π/2)Σ_(n=1) ^∞ (1/(n!))=(π/2)(e−1)

$${e}^{{cos}\left({x}\right)} {sin}\left({sinx}\right)=\Sigma\frac{{sin}\left({nx}\right)}{{n}!} \\ $$$$\int_{\mathrm{0}} ^{\infty} \underset{{n}=\mathrm{1}} {\sum}\frac{{sin}\left({nx}\right)}{{xn}!}=\frac{\pi}{\mathrm{2}}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}=\frac{\pi}{\mathrm{2}}\left({e}−\mathrm{1}\right) \\ $$

Commented by mnjuly1970 last updated on 15/May/21

mercey mr Payan...

$$\:\:\:\:\:{mercey}\:{mr}\:{Payan}... \\ $$

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