`int(e^(x))/(sqrt(4+e^(2x)))dx`

Explanation

To solve the integral \(\int \frac{e^x}{\sqrt{4 + e^{2x}}} \, dx\), we will use a substitution method. Here are the steps to solve the integral:

### Step 1: Substitution
Let \( t = e^x \). Then, the differential \( dt = e^x \, dx \) or \( dx = \frac{dt}{t} \).

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int \frac{e^x}{\sqrt{4 + e^{2x}}} \, dx = \int \frac{t}{\sqrt{4 + t^2}} \cdot \frac{dt}{t} = \int \frac{1}{\sqrt{4 + t^2}} \, dt \]

### Step 3: Identify the Integral Form
The integral \(\int \frac{1}{\sqrt{a^2 + x^2}} \, dx\) is a standard form, which equals \(\ln |x + \sqrt{x^2 + a^2}| + C\). Here, \( a^2 = 4 \) (thus \( a = 2 \)) and \( x = t \).

### Step 4: Evaluate the Integral
Using the standard form:
\[ \int \frac{1}{\sqrt{4 + t^2}} \, dt = \ln |t + \sqrt{t^2 + 4}| + C \]

### Step 5: Back Substitute
Now, we substitute back \( t = e^x \):
\[ \ln |e^x + \sqrt{(e^x)^2 + 4}| + C = \ln |e^x + \sqrt{e^{2x} + 4}| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{e^x}{\sqrt{4 + e^{2x}}} \, dx = \ln \left( e^x + \sqrt{e^{2x} + 4} \right) + C \]