Evaluate the following integrals:


Evaluate $\int \mathrm{e}^{2 \mathrm{x}}\left(\frac{1+\sin 2 \mathrm{x}}{1+\cos 2 \mathrm{x}}\right) \mathrm{dx}$


Put $2 x=t d x=d t / 2$

$\frac{1}{2} \int e^{t}\left(\frac{1+\sin t}{1+\cos t}\right) d t=\frac{1}{2} \int\left(e^{t} \tan \frac{t}{2}+\frac{1}{2} e^{t} \sec ^{2} \frac{t}{2}\right) d t$

$=\frac{1}{2} \int\left(e^{t} \tan \frac{t}{2}\right) d t+\frac{1}{4} \int \mathrm{e}^{\mathrm{t}} \sec ^{2} \frac{t}{2} d t$

$=\frac{1}{2} \int\left(e^{t} \tan \frac{t}{2}\right) d t+\frac{1}{4}\left[2 e^{t} \tan \frac{t}{2}-\int 2 e^{t} \tan \frac{t}{2}\right]=e^{t} \frac{\tan \frac{t}{2}}{2}+c$

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