Question Number 66114 by Rio Michael last updated on 09/Aug/19
$$\int\left(\frac{{e}^{\mathrm{2}{x}} −{sin}\mathrm{2}{x}}{{e}^{\mathrm{2}{x}} +{cos}\mathrm{2}{x}}\right){dx}\:=\:? \\ $$
Answered by $@ty@m123 last updated on 09/Aug/19
$${Let}\:{e}^{\mathrm{2}{x}} +\mathrm{cos}\:\mathrm{2}{x}={z} \\ $$$$\Rightarrow\:\left(\mathrm{2}{e}^{\mathrm{2}{x}} −\mathrm{2sin}\:\mathrm{2}{x}\right){dx}={dz} \\ $$$$\therefore\:{The}\:{given}\:{integral} \\ $$$$\int\frac{{dz}}{\mathrm{2}{z}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:{z}+{C} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left(\:{e}^{\mathrm{2}{x}} +\mathrm{cos}\:\mathrm{2}{x}\right)+{C} \\ $$$$ \\ $$
Commented by Rio Michael last updated on 09/Aug/19
$${i}\:{appreciate} \\ $$