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Author: Tinku Tara

1-0-pi-2-x-2-sin-2-x-dx-1-2-2-0-pi-2-x-tan-x-

Question Number 195846 by mnjuly1970 last updated on 11/Aug/23 $$ \\ $$$$\:\:\:\:\:\:\begin{cases}{\:\:\:\Omega_{\mathrm{1}} \:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{x}^{\:\mathrm{2}} }{{sin}^{\:\mathrm{2}} \left({x}\right)}\:{dx}\:}\\{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\frac{\Omega_{\mathrm{1}} }{\Omega_{\:\mathrm{2}} }\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}\\{\:\:\Omega_{\:\mathrm{2}} =\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{x}}{{tan}\left({x}\right)}\:{dx}}\end{cases} \\ $$$$ \\…

Question-195772

Question Number 195772 by dimentri last updated on 10/Aug/23 $$\:\:\:\:\cancel{\underline{\underbrace{ }}}\:\cancel{ } \\ $$ Answered by MM42 last updated on 10/Aug/23 $${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{3}}} \:\:\frac{{cos}\frac{\mathrm{3}{x}}{\mathrm{2}}\:−\mathrm{2}{cos}\frac{\mathrm{3}{x}}{\mathrm{2}}×{sin}\frac{\mathrm{3}{x}}{\mathrm{2}}}{\mathrm{4}{cos}\frac{\mathrm{3}{x}}{\mathrm{2}}{sin}\frac{\mathrm{3}{x}}{\mathrm{2}}{cos}\mathrm{3}{x}} \\ $$$$={lim}_{{x}\rightarrow\frac{\pi}{\mathrm{3}}}…