`int((1+cosx))/(x+sinx)dx`

Explanation

To solve the integral \( \int \frac{1 + \cos x}{x + \sin x} \, dx \), we can use the substitution method. Here’s the step-by-step solution:

### Step 1: Set Up the Integral
We start with the integral:
\[ I = \int \frac{1 + \cos x}{x + \sin x} \, dx \]

### Step 2: Choose a Substitution
We notice that the denominator \( x + \sin x \) can be a good candidate for substitution. Let:
\[ t = x + \sin x \]

### Step 3: Differentiate the Substitution
Next, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 1 + \cos x \]
This implies:
\[ dt = (1 + \cos x) \, dx \]

### Step 4: Substitute in the Integral
Now, we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{1 + \cos x} \]
Substituting \( t \) and \( dt \) into the integral gives:
\[ I = \int \frac{dt}{t} \]

### Step 5: Integrate
The integral \( \int \frac{dt}{t} \) is a standard integral:
\[ I = \ln |t| + C \]
where \( C \) is the constant of integration.

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

### Final Answer
Thus, the final answer is:
\[ \int \frac{1 + \cos x}{x + \sin x} \, dx = \ln |x + \sin x| + C \]

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