Question

\[\int\frac{e^{2x}}{e^{2x} - 2} dx\]

Answer 1

\[\text{Let I} = \int\frac{e^{2x}}{e^{2x} - 2}dx\]
\[\text{Putting }e^{2x} = t\]
\[ \Rightarrow 2 e^{2x} = \frac{dt}{dx}\]
\[ \Rightarrow e^{2x} dx = \frac{dt}{2}\]
\[ \therefore I = \frac{1}{2}\int\frac{1}{t - 2}dt\]
\[ = \frac{1}{2} \text{ln }\left| t - 2 \right| + C\]
\[ = \frac{1}{2} \text{ln }\left| e^{2x} - 2 \right| + C \left[ \because t = e^{2x} \right]\]