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Question-63750

Question Number 63750 by Tawa1 last updated on 08/Jul/19 Commented by Prithwish sen last updated on 08/Jul/19 $$\mathrm{perimeter}\:\mathrm{of}\:\mathrm{circle}\:\mathrm{A}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\pi\mathrm{r}\:\mathrm{unit} \\ $$$$\mathrm{perimeter}\:\mathrm{of}\:\mathrm{circle}\:\mathrm{B}\:=\:\mathrm{2}\pi\mathrm{r}\:\mathrm{unit} \\ $$$$\mathrm{Circle}\:\mathrm{A} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{covers}\:\frac{\mathrm{2}}{\mathrm{3}}\pi\mathrm{r}\:\mathrm{in}\:\mathrm{1}\:\mathrm{time} \\…

What-does-sin-2-x-mean-

Question Number 63747 by Tony Lin last updated on 08/Jul/19 $${What}\:{does}\:{sin}^{−\mathrm{2}} {x}\:{mean}? \\ $$ Commented by MJS last updated on 08/Jul/19 $$\mathrm{that}'\mathrm{s}\:\mathrm{the}\:\mathrm{question}… \\ $$$$\mathrm{we}'\mathrm{re}\:\mathrm{not}\:\mathrm{consequent}\:\mathrm{in}\:\mathrm{these}\:\mathrm{things} \\…

cos-2-x-cos-x-1-sin-x-cos-x-dx-

Question Number 129274 by bramlexs22 last updated on 14/Jan/21 $$\:\int\:\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:? \\ $$ Answered by liberty last updated on 14/Jan/21 $$\:\int\:\frac{\mathrm{cos}\:\mathrm{x}\left(\mathrm{cos}\:\mathrm{x}+\mathrm{1}\right)}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\:\mathrm{dx}\:=\:\int\:\frac{\mathrm{cos}\:\mathrm{x}\left(\mathrm{2cos}\:^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{2cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left[\:\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right]}\:\mathrm{dx}= \\ $$$$\:\int\:\frac{\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)\left(\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)}{\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{dx}=…

useful-formula-a-R-b-R-a-sin-x-b-cos-x-a-2-b-2-sin-x-arctan-b-a-dx-a-sin-x-b-cos-x-1-a-2-b-2-dx-sin-x-arctan-b-a-t-x-arctan-b-a-dx-dt-

Question Number 63738 by MJS last updated on 08/Jul/19 $$\mathrm{useful}\:\mathrm{formula} \\ $$$$======== \\ $$$$ \\ $$$$\forall{a}\in\mathbb{R}^{+} :\forall{b}\:\in\mathbb{R}:\:{a}\:\mathrm{sin}\:{x}\:+{b}\:\mathrm{cos}\:{x}\:=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\mathrm{sin}\:\left({x}+\mathrm{arctan}\:\frac{{b}}{{a}}\right) \\ $$$$\int\frac{{dx}}{{a}\:\mathrm{sin}\:{x}\:+{b}\:\mathrm{cos}\:{x}}= \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }}\int\frac{{dx}}{\mathrm{sin}\:\left({x}+\mathrm{arctan}\:\frac{{b}}{{a}}\right)}=…

nice-calculus-evsluate-n-0-1-n-n-3-2-2-n-2n-2-

Question Number 129272 by mnjuly1970 last updated on 14/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}… \\ $$$$\:\:\:{evsluate}:: \\ $$$$\:\:\:\:\:\Omega=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \:\left(\frac{\Gamma\left({n}+\frac{\mathrm{3}}{\mathrm{2}}\right)}{\mathrm{2}^{{n}} \:\Gamma\left(\:\mathrm{2}{n}\:+\mathrm{2}\right)}\right)=??? \\ $$$$ \\ $$ Answered by Dwaipayan…