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Question Number 151486 by mathocean1 last updated on 21/Aug/21

$$\int ln\left(sinx\right)=?$$$$x\ne 0+2k\pi ;k\in \mathbb{Z}$$

Answered by Olaf_Thorendsen last updated on 21/Aug/21

$$\mathrm{F}\left(x\right)=\int \ln \left(\sin x\right)dx$$$$\mathrm{F}\left(x\right)=x\ln \left(\sin x\right)-\int x\cot xdx$$$$\mathrm{F}\left(x\right)=x\ln \left(\sin x\right)-\int x\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{2}^{2n}{B}_{2n}{x}^{2n-1}}{\left(2n\right)!}dx$$$$\mathrm{F}\left(x\right)=x\ln \left(\sin x\right)-\int \underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{2}^{2n}{B}_{2n}{x}^{2n}}{\left(2n\right)!}dx$$$$\mathrm{F}\left(x\right)=x\ln \left(\sin x\right)-\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^n{2}^{2n}{B}_{2n}{x}^{2n+1}}{(2n+1)!}+\mathrm{C}$$

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