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Question Number 82391 by jagoll last updated on 21/Feb/20

∫ sin x cos (sin x) dx ?

$$\int\:\mathrm{sin}\:{x}\:\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)\:{dx}\:? \\ $$

Commented by M±th+et£s last updated on 21/Feb/20

i think that its special integral  cos(sin(x))=1−((sin^2 (x))/(2i))+((sin^4 (x))/(4i))−((sin^6 (x))/(6i))+...  I=∫(sin(x)−((sin^3 (x))/(2i))+((sin^5 (x))/(4i))−((sin^7 (x))/(7i))+...)dx

$${i}\:{think}\:{that}\:{its}\:{special}\:{integral} \\ $$$${cos}\left({sin}\left({x}\right)\right)=\mathrm{1}−\frac{{sin}^{\mathrm{2}} \left({x}\right)}{\mathrm{2}{i}}+\frac{{sin}^{\mathrm{4}} \left({x}\right)}{\mathrm{4}{i}}−\frac{{sin}^{\mathrm{6}} \left({x}\right)}{\mathrm{6}{i}}+... \\ $$$${I}=\int\left({sin}\left({x}\right)−\frac{{sin}^{\mathrm{3}} \left({x}\right)}{\mathrm{2}{i}}+\frac{{sin}^{\mathrm{5}} \left({x}\right)}{\mathrm{4}{i}}−\frac{{sin}^{\mathrm{7}} \left({x}\right)}{\mathrm{7}{i}}+...\right){dx} \\ $$

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