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Question Number 2176 by Filup last updated on 06/Nov/15
Is it possible to integrate the following:    ∫sin(cos θ)dθ
$$\mathrm{Is}\:\mathrm{it}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{integrate}\:\mathrm{the}\:\mathrm{following}: \\ $$$$ \\ $$$$\int\mathrm{sin}\left(\mathrm{cos}\:\theta\right){d}\theta \\ $$
Commented by 123456 last updated on 07/Nov/15
i dont know if it help but  sin x=Σ_(n=0) ^(+∞) (−1)^n (x^(2n+1) /((2n+1)!))=x−(x^3 /6)+∙∙∙  cos x=Σ_(n=0) ^(+∞) (−1)^n (x^(2n) /((2n)!))=1−(x^2 /2)+∙∙∙
$$\mathrm{i}\:\mathrm{dont}\:\mathrm{know}\:\mathrm{if}\:\mathrm{it}\:\mathrm{help}\:\mathrm{but} \\ $$$$\mathrm{sin}\:{x}=\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{{x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}+\centerdot\centerdot\centerdot \\ $$$$\mathrm{cos}\:{x}=\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \frac{{x}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}\right)!}=\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\centerdot\centerdot\centerdot \\ $$
Answered by prakash jain last updated on 07/Nov/15
you can do a series expansion of sin(cos θ)  and integrate.  Integral cannot be expressed in terms of  elementry functions alone.
$${you}\:{can}\:{do}\:{a}\:{series}\:{expansion}\:{of}\:\mathrm{sin}\left(\mathrm{cos}\:\theta\right) \\ $$$$\mathrm{and}\:\mathrm{integrate}. \\ $$$$\mathrm{Integral}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of} \\ $$$$\mathrm{elementry}\:\mathrm{functions}\:\mathrm{alone}. \\ $$
Commented by Filup last updated on 07/Nov/15
That makes sense. I wasn′t sure how it was  done. Thank you
$$\mathrm{That}\:\mathrm{makes}\:\mathrm{sense}.\:\mathrm{I}\:\mathrm{wasn}'\mathrm{t}\:\mathrm{sure}\:\mathrm{how}\:\mathrm{it}\:\mathrm{was} \\ $$$$\mathrm{done}.\:\mathrm{Thank}\:\mathrm{you} \\ $$

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