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Integration Questions

Question Number 45063 by rahul 19 last updated on 08/Oct/18

$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} {e}^{\mathrm{cos}\:\theta} \mathrm{cos}\:\left(\mathrm{sin}\:\theta\right){d}\theta\:=\:? \\$$

Commented by maxmathsup by imad last updated on 08/Oct/18

$${I}\:={Re}\left(\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{e}^{{cos}\theta+{isin}\theta} {d}\theta\right)\:\:{but}\:{e}^{{cos}\theta+{isin}\theta} \:={e}^{{e}^{{i}\theta} } \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left({e}^{{i}\theta} \right)^{{n}} }{{n}!} \\$$$$=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{{in}\theta} }{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({n}\theta\right)}{{n}!}\:+{i}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{sin}\left({n}\theta\right)}{{n}!}\:\Rightarrow \\$$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{e}^{{e}^{{i}\theta} } {d}\theta\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\int_{\mathrm{0}} ^{\mathrm{2}\pi} {cos}\left({n}\theta\right){d}\theta\:+{i}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{sin}\left({n}\theta\right){d}\theta \\$$$$=\mathrm{2}\pi\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\int_{\mathrm{0}} ^{\mathrm{2}\pi} {cos}\left({n}\theta\right){d}\theta\:+{i}\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{sin}\left({n}\theta\right){d}\theta\:{but} \\$$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} {cos}\left({n}\theta\right){d}\theta\:=\left[\frac{\mathrm{1}}{{n}}{sin}\left({n}\theta\right)\right]_{\mathrm{0}} ^{\mathrm{2}\pi} =\mathrm{0}\:\:\left({n}\geqslant\mathrm{1}\right)\:{also} \\$$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} {sin}\left({n}\theta\right){d}\theta\:=\left[−\frac{\mathrm{1}}{{n}}{cos}\left({n}\theta\right)\right]_{\mathrm{0}} ^{\mathrm{2}\pi} \:=\mathrm{0}\:\left({n}\geqslant\mathrm{1}\right)\:\Rightarrow\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:{e}^{{e}^{{i}\theta} } {d}\theta\:=\mathrm{2}\pi\:\Rightarrow \\$$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:{e}^{{cos}\theta} \:{cos}\left({sin}\theta\right){d}\theta\:=\mathrm{2}\pi\:. \\$$

Commented by tanmay.chaudhury50@gmail.com last updated on 08/Oct/18