Question Number 117811 by Lordose last updated on 13/Oct/20
$$\int_{\:\mathrm{0}} ^{\:\boldsymbol{\pi}} \boldsymbol{\mathrm{ln}}\mid\boldsymbol{\mathrm{sinh}}\left(\boldsymbol{\mathrm{x}}\right)\mid\boldsymbol{\mathrm{dx}} \\ $$
Answered by mindispower last updated on 13/Oct/20
$$=\int_{\mathrm{0}} ^{\pi} {ln}\left({sh}\left({x}\right)\right){dx} \\ $$$${sh}\left({x}\right)=\frac{{e}^{{x}} β{e}^{β{x}} }{\mathrm{2}} \\ $$$$=\int_{\mathrm{0}} ^{\pi} \left({ln}\left({e}^{{x}} \:\left(\mathrm{1}β{e}^{β\mathrm{2}{x}} \right)\right)β{ln}\left(\mathrm{2}\right)\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi} \left({x}+{ln}\left(\mathrm{1}β{e}^{β\mathrm{2}{x}} \right)β{ln}\left(\mathrm{2}\right)\right){dx} \\ $$$$=\left[\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} β{xln}\left(\mathrm{2}\right)\right]_{\mathrm{0}} ^{\pi} \:\underset{={A}} {\:}+\int_{\mathrm{0}} ^{\pi} \left({ln}\left(\mathrm{1}β{e}^{β\mathrm{2}{x}} \right){dx}\underset{={B}} {\right)} \\ $$$${ln}\left(\mathrm{1}β{e}^{β\mathrm{2}{x}} \right)=β\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{{e}^{β\mathrm{2}\left({n}+\mathrm{1}{x}\right.} }{{n}+\mathrm{1}} \\ $$$$\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}β{e}^{β\mathrm{2}{x}} \right){dx}=β\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{{n}+\mathrm{1}}\int_{\mathrm{0}} ^{\pi} {e}^{β\mathrm{2}\left({n}+\right){x}} {dx} \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{{e}^{β\mathrm{2}\left({n}+\mathrm{1}\right)\pi} β\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }=\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\left({e}^{β\mathrm{2}\pi} \right)^{{n}} }{{n}^{\mathrm{2}} }β\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} } \\ $$$${Li}_{{s}} \left({x}\right)=\underset{{k}\geqslant\mathrm{1}} {\sum}\frac{{x}^{{k}} }{{k}^{{s}} } \\ $$$${we}\:{get}=\frac{\mathrm{1}}{\mathrm{2}}\left({Li}_{\mathrm{2}} \left({e}^{β\mathrm{2}\pi} \right)β{Li}_{\mathrm{2}} \left(\mathrm{1}\right)\right)=\frac{\mathrm{1}}{\mathrm{2}}{Li}_{\mathrm{2}} \left({e}^{β\mathrm{2}\pi} \right)β\frac{\mathrm{1}}{\mathrm{2}}\zeta\left(\mathrm{2}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{Li}_{\mathrm{2}} \left({e}^{β\mathrm{2}\pi} \right)β\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$$${A}=\frac{\mathrm{1}}{\mathrm{2}}\pi^{\mathrm{2}} β\pi{ln}\left(\mathrm{2}\right) \\ $$$${A}+{B}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{12}}β\pi{ln}\left(\mathrm{2}\right)+{Li}_{\mathrm{2}} \left({e}^{β\mathrm{2}\pi} \right) \\ $$$$\int_{\mathrm{0}} ^{\pi} {ln}\left({sh}\left({x}\right)\right){dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{12}}β\pi{ln}\left(\mathrm{2}\right)+{Li}_{\mathrm{2}} \left({e}^{β\mathrm{2}\pi} \right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$