`int( x + sin x )/( 1+ cos x) dx` is equal to

Options

A. `log | 1+ cos x | +C`
B. ` log | x | sin x | + C`
C. `x - tan ""(x)/( 2)+c`
D. `x tan""(x)/( 2) +C`

Answer

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

### Step 1: Rewrite the Denominator
We can use the trigonometric identity:
\[ 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \]
Thus, we can rewrite the integral as:
\[ \int \frac{x + \sin x}{2 \cos^2\left(\frac{x}{2}\right)} \, dx \]

### Step 2: Split the Integral
Now we can separate the integral into two parts:
\[ \int \frac{x}{2 \cos^2\left(\frac{x}{2}\right)} \, dx + \int \frac{\sin x}{2 \cos^2\left(\frac{x}{2}\right)} \, dx \]

### Step 3: Simplify Each Integral
The first integral becomes:
\[ \frac{1}{2} \int x \sec^2\left(\frac{x}{2}\right) \, dx \]
The second integral can be rewritten using the identity \( \sin x = 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) \):
\[ \frac{1}{2} \int \frac{2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right)}{2 \cos^2\left(\frac{x}{2}\right)} \, dx = \int \tan\left(\frac{x}{2}\right) \, dx \]

### Step 4: Integrate the First Part Using Integration by Parts
For the integral \( \frac{1}{2} \int x \sec^2\left(\frac{x}{2}\right) \, dx \), we will use integration by parts:
Let \( u = x \) and \( dv = \sec^2\left(\frac{x}{2}\right) \, dx \).
Then, \( du = dx \) and \( v = 2 \tan\left(\frac{x}{2}\right) \).

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \frac{1}{2} \left( x \cdot 2 \tan\left(\frac{x}{2}\right) - \int 2 \tan\left(\frac{x}{2}\right) \, dx \right) \]

### Step 5: Simplify the Result
The integral of \( \tan\left(\frac{x}{2}\right) \) is:
\[ - \log\left|\cos\left(\frac{x}{2}\right)\right| + C \]
Thus, we can combine everything:
\[ \int \frac{x + \sin x}{1 + \cos x} \, dx = x \tan\left(\frac{x}{2}\right) - \log\left|\cos\left(\frac{x}{2}\right)\right| + C \]

### Final Result
The final result can be simplified to:
\[ x \tan\left(\frac{x}{2}\right) + C \]