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Question Number 142028 by mnjuly1970 last updated on 25/May/21

............Calculus......... ∫_0 ^( ∞) ((sin(sin(x)).e^(cos(x)) )/x)dx=??? ............m.n.....

$$\:\:\:\:\:\:\:\:............{Calculus}.........\: \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({sin}\left({x}\right)\right).{e}^{{cos}\left({x}\right)} }{{x}}{dx}=??? \\ $$$$\:............{m}.{n}..... \\ $$

Answered by mindispower last updated on 25/May/21

=Im∫_0 ^∞ (e^(cos(x)+isin(x)) /x) Im∫_0 ^∞ (e^e^(ix) /x)dx=Im∫_0 ^∞ Σ_(k≥0) (e^(ikx) /(k!x))dx =Σ_(k≥0) ∫_0 ^∞ ((sin(kx))/(x.k!))dx=Σ_(k≥0) (1/(k!))∫_0 ^∞ ((sin(kx))/(kx))d(kx) =Σ_(k≥0) (1/(k!)).∫_0 ^∞ ((sin(x))/x)dx=(π/2)Σ_(k≥0) (1/(k!))=(π/2)e

$$={Im}\int_{\mathrm{0}} ^{\infty} \frac{{e}^{{cos}\left({x}\right)+{isin}\left({x}\right)} }{{x}} \\ $$$${Im}\int_{\mathrm{0}} ^{\infty} \frac{{e}^{{e}^{{ix}} } }{{x}}{dx}={Im}\int_{\mathrm{0}} ^{\infty} \underset{{k}\geqslant\mathrm{0}} {\sum}\frac{{e}^{{ikx}} }{{k}!{x}}{dx} \\ $$$$=\underset{{k}\geqslant\mathrm{0}} {\sum}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({kx}\right)}{{x}.{k}!}{dx}=\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{{k}!}\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({kx}\right)}{{kx}}{d}\left({kx}\right) \\ $$$$=\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{{k}!}.\int_{\mathrm{0}} ^{\infty} \frac{{sin}\left({x}\right)}{{x}}{dx}=\frac{\pi}{\mathrm{2}}\underset{{k}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{{k}!}=\frac{\pi}{\mathrm{2}}{e} \\ $$

Commented by mnjuly1970 last updated on 25/May/21

zendeh bashid tashakor .thanks alot mr power ...

$${zendeh}\:{bashid}\:\:{tashakor}\:.{thanks}\:{alot} \\ $$$${mr}\:{power}\:... \\ $$

Commented by mindispower last updated on 25/May/21

withe pleasur sir thank you

$${withe}\:{pleasur}\:{sir}\:{thank}\:{you} \\ $$

Answered by Dwaipayan Shikari last updated on 25/May/21

e^(cosx) sin(sinx)=Σ_(n=0) ^∞ ((sin(nx))/(n!)) ∫_0 ^∞ Σ_(n=0) ^∞ ((sin(nx))/(xn!))dx=Σ_(n=0) ^∞ (1/(n!))∫_0 ^∞ ((sin(nx))/x)dx=((πe)/2)

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

Commented by mnjuly1970 last updated on 26/May/21

thanks alot...

$$\:\:{thanks}\:{alot}... \\ $$

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