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Question Number 143810 by Jamshidbek last updated on 18/Jun/21

∫cos(cosx)dx=?

$$\:\:\:\:\int\mathrm{cos}\left(\mathrm{cosx}\right)\mathrm{dx}=? \\ $$

Answered by mathmax by abdo last updated on 18/Jun/21

∫ cos(cosx)dx =∫ Σ_(n=0) ^∞ (((−1)^n cos^(2n) x)/((2n!))dx =Σ_(n=0) ^∞ (((−1)^n )/((2n)!))∫ cos^(2n) dx=Σ_(n=0) ^∞ (((−1)^n )/((2n)!))w_n W_n =∫ cos^(2n) dx(wallis generalise)

$$\int\:\mathrm{cos}\left(\mathrm{cosx}\right)\mathrm{dx}\:=\int\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{cos}^{\mathrm{2n}} \mathrm{x}}{\left(\mathrm{2n}!\right.}\mathrm{dx} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}\int\:\mathrm{cos}^{\mathrm{2n}} \:\mathrm{dx}=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{2n}\right)!}\mathrm{w}_{\mathrm{n}} \\ $$$$\mathrm{W}_{\mathrm{n}} =\int\:\mathrm{cos}^{\mathrm{2n}} \:\mathrm{dx}\left(\mathrm{wallis}\:\mathrm{generalise}\right) \\ $$

Answered by mathmax by abdo last updated on 19/Jun/21

∫ cos(cosx)dx =Re(∫ e^(icosx) dx) but ∫ e^(icosx) dx =∫Σ_(n=0) ^∞ ((i^n cos^n x)/(n!))dx =Σ_(n=0) ^∞ (i^n /(n!))∫ cos^n dx and ∫ cos^n dx=∫ (((e^(ix) +e^(−ix) )/2))^n dx =(1/2^n )∫ Σ_(k=0) ^n C_n ^k e^(ikx) (e^(−ix) )^(n−k) =(1/2^n )Σ_(k=0) ^n C_n ^k ∫ e^(ikx−i(n−k)x) dx =(1/2^n )Σ_(k=0) ^n C_n ^k ∫ e^((2ik−in)x) dx =(1/2^n )Σ_(k=0) ^n (C_n ^k /(2ik−in)) +k ⇒ ∫ e^(icosx) dx=Σ_(n=0) ^∞ (i^n /(n!))((1/2^n )Σ_(k=0) ^n (C_n ^k /(i(2k−n))))+k

$$\int\:\mathrm{cos}\left(\mathrm{cosx}\right)\mathrm{dx}\:=\mathrm{Re}\left(\int\:\mathrm{e}^{\mathrm{icosx}} \mathrm{dx}\right)\:\:\mathrm{but} \\ $$$$\int\:\mathrm{e}^{\mathrm{icosx}} \mathrm{dx}\:=\int\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{i}^{\mathrm{n}} \:\mathrm{cos}^{\mathrm{n}} \mathrm{x}}{\mathrm{n}!}\mathrm{dx}\:=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{i}^{\mathrm{n}} }{\mathrm{n}!}\int\:\:\mathrm{cos}^{\mathrm{n}} \:\mathrm{dx}\:\mathrm{and} \\ $$$$\int\:\mathrm{cos}^{\mathrm{n}} \mathrm{dx}=\int\:\left(\frac{\mathrm{e}^{\mathrm{ix}} \:+\mathrm{e}^{−\mathrm{ix}} }{\mathrm{2}}\right)^{\mathrm{n}} \:\mathrm{dx}\:=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\int\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{e}^{\mathrm{ikx}} \left(\mathrm{e}^{−\mathrm{ix}} \right)^{\mathrm{n}−\mathrm{k}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\int\:\mathrm{e}^{\mathrm{ikx}−\mathrm{i}\left(\mathrm{n}−\mathrm{k}\right)\mathrm{x}} \:\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\int\:\mathrm{e}^{\left(\mathrm{2ik}−\mathrm{in}\right)\mathrm{x}} \:\mathrm{dx}\:=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} }{\mathrm{2ik}−\mathrm{in}}\:+\mathrm{k}\:\Rightarrow \\ $$$$\int\:\mathrm{e}^{\mathrm{icosx}} \mathrm{dx}=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{i}^{\mathrm{n}} }{\mathrm{n}!}\left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} }{\mathrm{i}\left(\mathrm{2k}−\mathrm{n}\right)}\right)+\mathrm{k} \\ $$

Commented by mathmax by abdo last updated on 20/Jun/21

∫ e^(icosx) dx =Σ_(n=0) ^∞ (i^n /(n!2^n ))Σ_(k=0) ^n (C_n ^k /(i(2k−n)))e^((2k−n)ix) +K

$$\int\:\mathrm{e}^{\mathrm{icosx}} \mathrm{dx}\:=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{i}^{\mathrm{n}} }{\mathrm{n}!\mathrm{2}^{\mathrm{n}} }\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} }{\mathrm{i}\left(\mathrm{2k}−\mathrm{n}\right)}\mathrm{e}^{\left(\mathrm{2k}−\mathrm{n}\right)\mathrm{ix}} \:+\mathrm{K} \\ $$

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