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

Answer

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

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

### Step 2: Use substitution
We notice that the derivative of the denominator \( e^x + \sin x \) resembles the numerator. Thus, we can use the substitution:
\[ t = e^x + \sin x \]
Now, differentiate both sides with respect to \( x \):
\[ \frac{dt}{dx} = e^x + \cos x \]
This implies:
\[ dt = (e^x + \cos x) \, dx \]

### Step 3: Rewrite the integral
Now we can rewrite the integral in terms of \( t \):
\[ I = \int \frac{dt}{t} \]

### Step 4: Integrate
The integral \( \int \frac{dt}{t} \) is a standard integral:
\[ I = \ln |t| + C \]
where \( C \) is the constant of integration.

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

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

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