Integration Questions

Question Number 207582 by sniper237 last updated on 19/May/24

$$\int_{\mathrm{0}} ^{\pi} \:{ln}\left({sinx}\right){dx}=−\pi{ln}\mathrm{2} \\$$$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\Gamma\left({x}\right){dx}\:=\:{ln}\left(\mathrm{2}\pi\right) \\$$

Answered by mathzup last updated on 21/May/24

$${I}=\int_{\mathrm{0}} ^{\pi} {ln}\left({sinx}\right){dx}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({sinx}\right){dx} \\$$$$+\int_{\frac{\pi}{\mathrm{2}}} ^{\pi} {ln}\left({sinx}\right){dx}\left(\rightarrow{x}=\frac{\pi}{\mathrm{2}}+{t}\right) \\$$$$=−\frac{\pi}{\mathrm{2}}{ln}\left(\mathrm{2}\right)+\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cost}\right){dt} \\$$$$=−\frac{\pi}{\mathrm{2}}{ln}\mathrm{2}−\frac{\pi}{\mathrm{2}}{ln}\mathrm{2}=−\pi{ln}\mathrm{2} \\$$

Answered by mathzup last updated on 21/May/24

$$\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\Rightarrow \\$$$${ln}\left(\Gamma\left({x}\right)\right)+{ln}\left(\Gamma\left(\mathrm{1}−{x}\right)\right)={ln}\pi−{ln}\left({sin}\left(\pi{x}\right)\right) \\$$$$\Rightarrow\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){dx}+\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left(\mathrm{1}−{x}\right)\right){dx} \\$$$$={ln}\left(\pi\right)−\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({sin}\left(\pi{x}\right)\right){dx}\left(\rightarrow\pi{x}={t}\right) \\$$$$={ln}\left(\pi\right)−\int_{\mathrm{0}} ^{\pi} {ln}\left({sint}\right)\frac{{dt}}{\pi} \\$$$$={ln}\left(\pi\right)−\frac{\mathrm{1}}{\pi}\left(−\pi{ln}\mathrm{2}\right) \\$$$$={ln}\left(\pi\right)+{ln}\left(\mathrm{2}\right)={ln}\left(\mathrm{2}\pi\right) \\$$$${but}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left(\mathrm{1}−{x}\right)\right){dx}=_{\mathrm{1}−{x}={t}} \int_{\mathrm{1}} ^{\mathrm{0}} {ln}\left(\Gamma\left({t}\right)\right)\left(−{dt}\right) \\$$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left({x}\right){dx}\:\Rightarrow\right. \\$$$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){dx}=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{2}\pi\right) \\$$$${il}\:{ya}\:{erreur}\:{dans}\:{l}\:{enonce}\:!... \\$$