Category: Logarithms

Question Number 197388 by cortano12 last updated on 16/Sep/23 Answered by MM42 last updated on 16/Sep/23 $$x=1\mathrm{\&}y=1$$ Answered by Rasheed.Sindhi last updated on…

Question Number 197099 by Erico last updated on 07/Sep/23 $${\underset{0}{\int }}^{\frac{\pi }{2}}\frac{\ln \left(\cos t\right)}{\sin t}\mathrm{d}t=???$$ Answered by witcher3 last updated on 07/Sep/23 $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\left(\mathrm{t}\right)\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{t}\right)\right)}{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \left(\mathrm{t}\right)}\mathrm{dt},\mathrm{cos}\left(\mathrm{t}\right)=\mathrm{y}…

Question Number 196683 by MATHEMATICSAM last updated on 29/Aug/23 $$\mathrm{If}y=\frac{e^x-e^{-x}}{e^x+e^{-x}}\mathrm{then}\mathrm{show}\mathrm{that}$$$$x=\frac{1}{2}{\log }_e\left(\frac{1+y}{1-y}\right).$$ Answered by Frix last updated on…

Question Number 196427 by Erico last updated on 24/Aug/23 $$\mathrm{If}f\left(x\right)={\underset{0}{\int }}^x\frac{dt}{t+e^{-f\left(t\right)}},\mathrm{determine}f\left(x\right)$$ Answered by witcher3 last updated on 25/Aug/23 $$\mathrm{f}\left(0\right)=0$$$$\mathrm{f}'\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{x}+\mathrm{e}^{−\mathrm{f}\left(\mathrm{x}\right)}…

Question Number 196026 by Erico last updated on 16/Aug/23 $${\underset{0}{\int }}^{+\mathrm{\infty }}\frac{{\left(lnt\right)}^2}{1+t^2}dt$$ Answered by sniper237 last updated on 16/Aug/23 $$=\:{f}''\left(\mathrm{0}\right)\:\:\:{with}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty}…