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Show-that-0-pi-2-1-cos-3-sin-2-d-1-3-

Question Number 153737 by ZiYangLee last updated on 09/Sep/21 $$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\left(\mathrm{cos}\:\theta+\:\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta\right)^{\mathrm{2}} }\:{d}\theta=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}\:} \\ $$ Answered by puissant last updated on 09/Sep/21 $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\left[\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}{cos}\theta+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{sin}\theta\right)\right]^{\mathrm{2}}…

Question-153696

Question Number 153696 by SANOGO last updated on 09/Sep/21 Answered by puissant last updated on 09/Sep/21 $${x}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +{k}} \\ $$$${Calcul}\:{des}\:\mathrm{5}\:{premiers}\:{termes}.. \\ $$$$\rightarrow\:{x}_{\mathrm{1}}…

solve-in-R-x-3-y-3-19-xy-6-

Question Number 153698 by mathocean1 last updated on 09/Sep/21 $${solve}\:{in}\:\mathbb{R} \\ $$$$\begin{cases}{{x}^{\mathrm{3}} −{y}^{\mathrm{3}} =\mathrm{19}}\\{{xy}=\mathrm{6}}\end{cases} \\ $$ Answered by amin96 last updated on 09/Sep/21 $$\left({x}−{y}\right)\left({x}^{\mathrm{2}} +{xy}+{y}^{\mathrm{2}}…

Question-153676

Question Number 153676 by SANOGO last updated on 09/Sep/21 Answered by som(math1967) last updated on 09/Sep/21 $$\mathrm{1}.\:\boldsymbol{{a}}>\mathrm{0},\boldsymbol{{b}}>\mathrm{0} \\ $$$$\:\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)^{\mathrm{2}} \geqslant\mathrm{0}\:\:\:\left[\boldsymbol{{if}}\:\boldsymbol{{a}}=\boldsymbol{{b}}\:\boldsymbol{{then}}\left(\boldsymbol{{a}}−\boldsymbol{{b}}\right)=\mathrm{0}\right] \\ $$$$\Rightarrow\left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)^{\mathrm{2}} −\mathrm{4}\boldsymbol{{ab}}\geqslant\mathrm{0} \\ $$$$\Rightarrow\:\left(\frac{\boldsymbol{{a}}+\boldsymbol{{b}}}{\mathrm{2}}\right)^{\mathrm{2}}…

Question-22567

Question Number 22567 by mondodotto@gmail.com last updated on 20/Oct/17 Answered by Rasheed.Sindhi last updated on 20/Oct/17 $$\mathrm{840}=\mathrm{2}^{\mathrm{3}} .\mathrm{3}.\mathrm{5}.\mathrm{7} \\ $$$$\therefore\:\mathrm{The}\:\mathrm{numbers}\:\left(\mathrm{of}\:\mathrm{which}\:\mathrm{840}\:\mathrm{is}\:\mathrm{lcm}\right) \\ $$$$\mathrm{may}\:\mathrm{have}\:\mathrm{factors}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7}\:\mathrm{all}\:\mathrm{or}\:\mathrm{some} \\ $$$$\mathrm{of}\:\mathrm{them}. \\…