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Question Number 149150 by mathdanisur last updated on 03/Aug/21

$$\int ln\left(cosx\right)dx=?$$

Answered by puissant last updated on 03/Aug/21

$$\mathrm{K}=\mathrm{xln}\left(\mathrm{cosx}\right)+\int \mathrm{xtanxdx}$$$$=\mathrm{xln}\left(\mathrm{cosx}\right)+\int \mathrm{x}\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\mid {\mathrm{B}}_{2\mathrm{n}}\mid \frac{2^{2\mathrm{n}}(2^{2\mathrm{n}}-1){\mathrm{x}}^{2\mathrm{n}-1}}{\left(2\mathrm{n}\right)!}\mathrm{dx}$$$$=\mathrm{xln}\left(\mathrm{cosx}\right)+\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\mid {\mathrm{B}}_{2\mathrm{n}}\mid \frac{2^{2\mathrm{n}}(2^{2\mathrm{n}}-1)}{\left(2\mathrm{n}\right)!}\int {\mathrm{x}}^{2\mathrm{n}}\mathrm{dx}$$$$\mathrm{soit}:$$$$\mathrm{I}=\mathrm{xln}\left(\mathrm{cosx}\right)+\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\mid {\mathrm{B}}_{2\mathrm{n}}\mid \frac{2^{2\mathrm{n}}(2^{2\mathrm{n}}-1)}{(2\mathrm{n}+1)!}{\mathrm{x}}^{2\mathrm{n}+1}+\mathrm{C}$$$$$$$$......\mathrm{Le}\mathrm{puissant}...$$

Commented by mathdanisur last updated on 03/Aug/21

$$ThankYouSer$$

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