`int e^(2x)*{(1+sin 2x)/(1+cos 2x)}dx`

Answer

To solve the integral \( \int e^{2x} \cdot \frac{1 + \sin(2x)}{1 + \cos(2x)} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( u = 2x \). Then, \( du = 2 \, dx \) or \( dx = \frac{du}{2} \).

### Step 2: Rewrite the Integral
Substituting \( u \) into the integral gives us:
\[ \int e^{2x} \cdot \frac{1 + \sin(2x)}{1 + \cos(2x)} \, dx = \int e^u \cdot \frac{1 + \sin(u)}{1 + \cos(u)} \cdot \frac{du}{2} \]
This simplifies to:
\[ \frac{1}{2} \int e^u \cdot \frac{1 + \sin(u)}{1 + \cos(u)} \, du \]

### Step 3: Split the Integral
Now, we can split the integral into two parts:
\[ \frac{1}{2} \left( \int e^u \cdot \frac{1}{1 + \cos(u)} \, du + \int e^u \cdot \frac{\sin(u)}{1 + \cos(u)} \, du \right) \]

### Step 4: Simplify the First Integral
Using the identity \( 1 + \cos(u) = 2 \cos^2\left(\frac{u}{2}\right) \), we can rewrite the first integral:
\[ \int e^u \cdot \frac{1}{1 + \cos(u)} \, du = \int e^u \cdot \frac{1}{2 \cos^2\left(\frac{u}{2}\right)} \, du = \frac{1}{2} \int e^u \sec^2\left(\frac{u}{2}\right) \, du \]

### Step 5: Solve the Second Integral
For the second integral, we can use the identity \( \sin(u) = 2 \sin\left(\frac{u}{2}\right) \cos\left(\frac{u}{2}\right) \):
\[ \int e^u \cdot \frac{\sin(u)}{1 + \cos(u)} \, du = \int e^u \cdot \frac{2 \sin\left(\frac{u}{2}\right) \cos\left(\frac{u}{2}\right)}{1 + \cos(u)} \, du \]
This simplifies to:
\[ \int e^u \cdot \frac{2 \sin\left(\frac{u}{2}\right) \cos\left(\frac{u}{2}\right)}{2 \cos^2\left(\frac{u}{2}\right)} \, du = \int e^u \cdot \tan\left(\frac{u}{2}\right) \, du \]

### Step 6: Integration by Parts
Now we can use integration by parts on both integrals:
1. For \( \int e^u \sec^2\left(\frac{u}{2}\right) \, du \), let \( v = e^u \) and \( dw = \sec^2\left(\frac{u}{2}\right) \, du \).
2. For \( \int e^u \tan\left(\frac{u}{2}\right) \, du \), let \( v = e^u \) and \( dw = \tan\left(\frac{u}{2}\right) \, du \).

### Step 7: Combine Results
After performing integration by parts, we will combine the results from both integrals and simplify.

### Step 8: Back Substitution
Finally, substitute back \( u = 2x \) to express the result in terms of \( x \).

### Final Answer
The final result of the integral is:
\[ \frac{1}{2} e^{2x} \tan(x) + C \]
where \( C \) is the constant of integration.