`int(e^(x)+cos x)/(e^(x) +sin x) dx`

Answer

To solve the integral \( \int \frac{e^x + \cos x}{e^x + \sin x} \, dx \), we will use substitution methods. Here’s a step-by-step solution:

### Step 1: Set up the integral
Let
\[ I = \int \frac{e^x + \cos x}{e^x + \sin x} \, dx \]

### Step 2: Choose a substitution
We will make the substitution:
\[ t = e^x + \sin x \]

### Step 3: Differentiate the substitution
Now, we differentiate \( t \) with respect to \( x \):
\[ dt = e^x \, dx + \cos x \, dx = (e^x + \cos x) \, dx \]
This implies:
\[ dx = \frac{dt}{e^x + \cos x} \]

### Step 4: Substitute in the integral
Substituting \( t \) and \( dx \) into the integral, we have:
\[ I = \int \frac{e^x + \cos x}{t} \cdot \frac{dt}{e^x + \cos x} \]
The \( e^x + \cos x \) terms cancel out:
\[ I = \int \frac{1}{t} \, dt \]

### Step 5: Integrate
The integral of \( \frac{1}{t} \) is:
\[ I = \ln |t| + C \]

### Step 6: Substitute back for \( t \)
Now, we substitute back the value of \( t \):
\[ I = \ln |e^x + \sin x| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{e^x + \cos x}{e^x + \sin x} \, dx = \ln |e^x + \sin x| + C \]