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

Answer

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

### Step 1: Substitution
Let \( t = 1 + \sin x \). Then, the derivative of \( t \) with respect to \( x \) is:
\[ \frac{dt}{dx} = \cos x \implies dt = \cos x \, dx \]
This means we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{\cos x} \]

### Step 2: Rewrite the Integral
Substituting \( t \) and \( dx \) into the integral, we have:
\[ \int \frac{\cos x}{1 + \sin x} \, dx = \int \frac{\cos x}{t} \cdot \frac{dt}{\cos x} \]
The \( \cos x \) terms cancel out:
\[ = \int \frac{dt}{t} \]

### Step 3: Integrate
The integral \( \int \frac{dt}{t} \) is a standard integral:
\[ = \log |t| + C \]

### Step 4: Substitute Back
Now, substitute back \( t = 1 + \sin x \):
\[ = \log |1 + \sin x| + C \]

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