`int(cosx)/((1+cosx))dx=?`

Options

A. `x+"tan"(x)/(2)+C`
B. `-x+"tan"(x)/(2)+C`
C. `x-"tan"(x)/(2)+C`
D. none of these

Explanation

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

### Step 1: Rewrite the Integral
We start by rewriting the integral:
\[ \int \frac{\cos x}{1 + \cos x} \, dx \]
We can manipulate the integrand by adding and subtracting 1 in the numerator:
\[ \int \frac{\cos x + 1 - 1}{1 + \cos x} \, dx = \int \frac{(1 + \cos x) - 1}{1 + \cos x} \, dx \]
This simplifies to:
\[ \int \left( 1 - \frac{1}{1 + \cos x} \right) \, dx \]

### Step 2: Split the Integral
Now we can split the integral into two separate integrals:
\[ \int 1 \, dx - \int \frac{1}{1 + \cos x} \, dx \]
The first integral is straightforward:
\[ \int 1 \, dx = x \]

### Step 3: Simplify the Second Integral
Next, we focus on the second integral:
\[ \int \frac{1}{1 + \cos x} \, dx \]
We can use the identity \( \cos x = \frac{1 - \tan^2(\frac{x}{2})}{1 + \tan^2(\frac{x}{2})} \) to rewrite \( 1 + \cos x \):
\[ 1 + \cos x = 1 + \frac{1 - \tan^2(\frac{x}{2})}{1 + \tan^2(\frac{x}{2})} = \frac{2}{1 + \tan^2(\frac{x}{2})} \]
Thus,
\[ \frac{1}{1 + \cos x} = \frac{1 + \tan^2(\frac{x}{2})}{2} \]
Now we can rewrite the integral:
\[ \int \frac{1}{1 + \cos x} \, dx = \frac{1}{2} \int (1 + \tan^2(\frac{x}{2})) \, dx \]

### Step 4: Integrate
The integral of \( 1 \) is \( x \), and the integral of \( \tan^2(\frac{x}{2}) \) can be computed using the identity \( \tan^2 t = \sec^2 t - 1 \):
\[ \int \tan^2(\frac{x}{2}) \, dx = \int (\sec^2(\frac{x}{2}) - 1) \, dx = 2 \tan(\frac{x}{2}) - x \]
Thus,
\[ \int (1 + \tan^2(\frac{x}{2})) \, dx = x + 2 \tan(\frac{x}{2}) - x = 2 \tan(\frac{x}{2}) \]
So,
\[ \int \frac{1}{1 + \cos x} \, dx = \frac{1}{2} (x + 2 \tan(\frac{x}{2})) = \frac{x}{2} + \tan(\frac{x}{2}) \]

### Step 5: Combine Results
Now we combine the results from Step 2:
\[ \int \frac{\cos x}{1 + \cos x} \, dx = x - \left( \frac{x}{2} + \tan(\frac{x}{2}) \right) \]
This simplifies to:
\[ \frac{x}{2} - \tan(\frac{x}{2}) + C \]

### Final Answer
Thus, the final result is:
\[ \int \frac{\cos x}{1 + \cos x} \, dx = \frac{x}{2} - \tan\left(\frac{x}{2}\right) + C \]