Question Number 208075 by hardmath last updated on 04/Jun/24
$$\mathrm{Find}:\:\:\:\int\:\mathrm{sin}\:\left(\mathrm{2x}\:−\:\frac{\pi}{\mathrm{4}}\right)\:=\:? \\ $$
Answered by TonyCWX08 last updated on 04/Jun/24
$${sin}\left(\mathrm{2}{x}−\frac{\pi}{\mathrm{4}}\right) \\ $$$$={sin}\left(\mathrm{2}{x}\right){cos}\left(\frac{\pi}{\mathrm{4}}\right)−{sin}\left(\frac{\pi}{\mathrm{4}}\right){cos}\left(\mathrm{2}{x}\right) \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\left({sin}\left(\mathrm{2}{x}\right)−{cos}\left(\mathrm{2}{x}\right)\right) \\ $$$$ \\ $$$$\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\int\:{sin}\left(\mathrm{2}{x}\right)−{cos}\left(\mathrm{2}{x}\right)\:{dx} \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\left(−\frac{{cos}\left(\mathrm{2}{x}\right)+{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}}\right)+{c} \\ $$$$=−\frac{\sqrt{\mathrm{2}}\left({cos}\left({x}\right)+{sin}\left({x}\right)\right)}{\mathrm{4}}+{c} \\ $$