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Author: Tinku Tara

sin-x-cos-3x-sin-x-cos-2x-dx-

Question Number 85718 by M±th+et£s last updated on 24/Mar/20 $$\int\frac{{sin}\left({x}\right)−{cos}\left(\mathrm{3}{x}\right)}{{sin}\left({x}\right)−{cos}\left(\mathrm{2}{x}\right)}{dx} \\ $$ Answered by MJS last updated on 24/Mar/20 $$\int\frac{\mathrm{sin}\:{x}\:−\mathrm{cos}\:\mathrm{3}{x}}{\mathrm{sin}\:{x}\:−\mathrm{cos}\:\mathrm{2}{x}}{dx}= \\ $$$$=\int\frac{\mathrm{4cos}\:{x}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:−\mathrm{cos}\:{x}\:+\mathrm{sin}\:{x}}{−\mathrm{2cos}^{\mathrm{2}} \:{x}\:+\mathrm{sin}\:{x}\:+\mathrm{1}}{dx}= \\…

Question-151248

Question Number 151248 by mathdanisur last updated on 19/Aug/21 Answered by EDWIN88 last updated on 19/Aug/21 $$\mathrm{4sin}\:\left(\mathrm{4}{x}−\mathrm{60}°\right)\mathrm{sin}\:\left(\mathrm{6}{x}−\mathrm{60}°\right)\mathrm{sin}\left(\mathrm{480}°−\mathrm{10}{x}\right)+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{0} \\ $$$$\left\{\mathrm{2sin}\:\left(\mathrm{4}{x}−\mathrm{60}°\right)\mathrm{sin}\:\left(\mathrm{6}{x}−\mathrm{60}°\right)\right\}\mathrm{2sin}\:\left(\mathrm{120}°−\mathrm{10}{x}\right)+\mathrm{sin}\:\mathrm{60}°=\mathrm{0} \\ $$$$\left\{\mathrm{cos}\:\mathrm{2}{x}−\mathrm{cos}\:\left(\mathrm{10}{x}−\mathrm{120}°\right)\right\}\mathrm{2sin}\:\left(\mathrm{120}°−\mathrm{10}{x}\right)+\mathrm{sin}\:\mathrm{60}°=\mathrm{0} \\ $$$$\mathrm{2cos}\:\mathrm{2}{x}\:\mathrm{sin}\:\left(\mathrm{120}°−\mathrm{10}{x}\right)−\mathrm{sin}\:\left(\mathrm{240}°−\mathrm{20}{x}\right)+\mathrm{sin}\:\mathrm{60}°=\mathrm{0} \\ $$$$\mathrm{sin}\:\left(\mathrm{120}°−\mathrm{8}{x}\right)−\mathrm{sin}\:\left(\mathrm{12}{x}−\mathrm{120}°\right)+\mathrm{sin}\:\left(\mathrm{60}°−\mathrm{20}{x}\right)+\mathrm{sin}\:\mathrm{60}°=\mathrm{0}…

0-1-2-1-9-0-16-2-

Question Number 20174 by DKumar last updated on 23/Aug/17 $$\left(\mathrm{0}.\overset{−} {\mathrm{1}}\right)^{\mathrm{2}} \left\{\mathrm{1}−\mathrm{9}\left(\mathrm{0}.\mathrm{1}\overset{−} {\mathrm{6}}\right)^{\mathrm{2}} \right\} \\ $$ Answered by mrW1 last updated on 23/Aug/17 $$\mathrm{0}.\overline {\mathrm{1}}=\mathrm{0}.\mathrm{111111}…=\frac{\mathrm{1}}{\mathrm{9}}…

1-5-1-5-2-2-1-5-3-3-1-5-4-4-

Question Number 151247 by mathdanisur last updated on 19/Aug/21 $$\frac{\mathrm{1}}{\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} \centerdot\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} \centerdot\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{4}} \centerdot\mathrm{4}}\:+\:…\:=\:? \\ $$ Answered by qaz last updated on 19/Aug/21 $$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n5}^{\mathrm{n}}…