Evaluate : `int(e^(2x))/(e^(x) + 2) dx`

Answer

To evaluate the integral \( I = \int \frac{e^{2x}}{e^x + 2} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = e^x + 2 \). Then, we need to find \( dt \):
\[ dt = e^x \, dx \quad \Rightarrow \quad dx = \frac{dt}{e^x} \]
From our substitution, we can express \( e^x \) in terms of \( t \):
\[ e^x = t - 2 \]

### Step 2: Rewrite the Integral
Substituting \( e^x \) and \( dx \) into the integral:
\[ I = \int \frac{e^{2x}}{e^x + 2} \, dx = \int \frac{(e^x)^2}{e^x + 2} \, dx = \int \frac{(t - 2)^2}{t} \cdot \frac{dt}{t - 2} \]
Now, substituting \( e^x = t - 2 \):
\[ I = \int \frac{(t - 2)^2}{t} \cdot \frac{dt}{t - 2} = \int \frac{(t - 2)}{t} \, dt \]

### Step 3: Simplify the Integral
Now, we can simplify the integrand:
\[ \frac{(t - 2)}{t} = 1 - \frac{2}{t} \]
Thus, the integral becomes:
\[ I = \int \left( 1 - \frac{2}{t} \right) dt \]

### Step 4: Integrate
Now we can integrate term by term:
\[ I = \int 1 \, dt - 2 \int \frac{1}{t} \, dt = t - 2 \ln |t| + C \]

### Step 5: Substitute Back
Now we substitute back \( t = e^x + 2 \):
\[ I = (e^x + 2) - 2 \ln |e^x + 2| + C \]

### Final Answer
Thus, the evaluated integral is:
\[ I = e^x + 2 - 2 \ln(e^x + 2) + C \]