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

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

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

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

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

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

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

$$\int {\mathrm{e}}^{\mathrm{icosx}}\mathrm{dx}=\sum _{\mathrm{n}=0}^{\mathrm{\infty }}\frac{{\mathrm{i}}^{\mathrm{n}}}{\mathrm{n}!2^{\mathrm{n}}}\sum _{\mathrm{k}=0}^{\mathrm{n}}\frac{{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}}{\mathrm{i}(2\mathrm{k}-\mathrm{n})}{\mathrm{e}}^{(2\mathrm{k}-\mathrm{n})\mathrm{ix}}+\mathrm{K}$$

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