Question:

Put, $xe^{2x} = t \Rightarrow (e^{2x} + 2xe^{2x}) dx = dt \Rightarrow e^{2x}(1+2x) dx = dt$<br>$\therefore \int \frac{e^{2x}(1+2x)}{\sin(xe^{2x})} dx = \int \frac{dt}{\sin t} = \int \text{cosec} t \, dt = \ln \left| \tan \frac{t}{2} \right| + C = \ln \left| \tan \frac{xe^{2x}}{2} \right| + C$