`inte^((x)/(2))((2-sinx)/(1-cosx))dx=`

Options

A. `2e^((x)/(2))cot((x)/(2))+c`
B. `-2e^((x)/(2))cot((x)/(2))+c`
C. `e^((x)/(2))cot((x)/(2))+c`
D. `-e^((x)/(2))cot((x)/(2))+c`

Answer

To solve the integral \( I = \int e^{\frac{x}{2}} \frac{2 - \sin x}{1 - \cos x} \, dx \), we will follow these steps:

### Step 1: Substitute for \( x \)
Let \( t = \frac{x}{2} \). Then, \( x = 2t \) and \( dx = 2 \, dt \).

### Step 2: Rewrite the integral
Substituting \( x \) and \( dx \) into the integral, we have:
\[ I = \int e^{t} \frac{2 - \sin(2t)}{1 - \cos(2t)} \cdot 2 \, dt = 2 \int e^{t} \frac{2 - \sin(2t)}{1 - \cos(2t)} \, dt \]

### Step 3: Simplify the expression
Using the double angle identities:
- \( \sin(2t) = 2 \sin(t) \cos(t) \)
- \( \cos(2t) = 1 - 2 \sin^2(t) \)

Thus, \( 1 - \cos(2t) = 2 \sin^2(t) \). Now, substituting these into the integral gives:
\[ I = 2 \int e^{t} \frac{2 - 2 \sin(t) \cos(t)}{2 \sin^2(t)} \, dt \]
This simplifies to:
\[ I = 2 \int e^{t} \left( \frac{2(1 - \sin(t) \cos(t))}{2 \sin^2(t)} \right) \, dt = 2 \int e^{t} \left( \frac{1 - \sin(t) \cos(t)}{\sin^2(t)} \right) \, dt \]

### Step 4: Separate the integral
We can separate the integral into two parts:
\[ I = 2 \int e^{t} \frac{1}{\sin^2(t)} \, dt - 2 \int e^{t} \frac{\sin(t) \cos(t)}{\sin^2(t)} \, dt \]
This simplifies to:
\[ I = 2 \int e^{t} \csc^2(t) \, dt - 2 \int e^{t} \cot(t) \, dt \]

### Step 5: Solve the integrals
1. The integral \( \int e^{t} \csc^2(t) \, dt \) can be solved using integration by parts or known results.
2. The integral \( \int e^{t} \cot(t) \, dt \) can also be solved using integration by parts.

### Step 6: Combine results
After evaluating both integrals, we combine the results and substitute back \( t = \frac{x}{2} \).

### Final Answer
The final result will be:
\[ I = -2 e^{\frac{x}{2}} \cot\left(\frac{x}{2}\right) + C \]