Question Number 66114 by Rio Michael last updated on 09/Aug/19
![∫(((e^(2x) −sin2x)/(e^(2x) +cos2x)))dx = ?](https://www.tinkutara.com/question/Q66114.png)
$$\int\left(\frac{{e}^{\mathrm{2}{x}} −{sin}\mathrm{2}{x}}{{e}^{\mathrm{2}{x}} +{cos}\mathrm{2}{x}}\right){dx}\:=\:? \\ $$
Answered by $@ty@m123 last updated on 09/Aug/19
![Let e^(2x) +cos 2x=z ⇒ (2e^(2x) −2sin 2x)dx=dz ∴ The given integral ∫(dz/(2z)) =(1/2)ln z+C =(1/2)ln ( e^(2x) +cos 2x)+C](https://www.tinkutara.com/question/Q66120.png)
$${Let}\:{e}^{\mathrm{2}{x}} +\mathrm{cos}\:\mathrm{2}{x}={z} \\ $$$$\Rightarrow\:\left(\mathrm{2}{e}^{\mathrm{2}{x}} −\mathrm{2sin}\:\mathrm{2}{x}\right){dx}={dz} \\ $$$$\therefore\:{The}\:{given}\:{integral} \\ $$$$\int\frac{{dz}}{\mathrm{2}{z}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:{z}+{C} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left(\:{e}^{\mathrm{2}{x}} +\mathrm{cos}\:\mathrm{2}{x}\right)+{C} \\ $$$$ \\ $$
Commented by Rio Michael last updated on 09/Aug/19
![i appreciate](https://www.tinkutara.com/question/Q66132.png)
$${i}\:{appreciate} \\ $$