Evaluate : `int (sqrt(1 + sin x))/( 1 + cos x) e^(-(x)/(2))dx`

Answer

To evaluate the integral

\[ I = \int \frac{\sqrt{1 + \sin x}}{1 + \cos x} e^{-\frac{x}{2}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the integrand
We start by rewriting \(\sqrt{1 + \sin x}\) and \(1 + \cos x\) using trigonometric identities.

Using the identities:
\[ 1 + \sin x = \cos^2\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) + 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) = \left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)^2, \]
and
\[ 1 + \cos x = 2\cos^2\left(\frac{x}{2}\right), \]
we can rewrite the integral as:

\[ I = \int \frac{\left(\cos\left(\frac{x}{2}\right) + \sin\left(\frac{x}{2}\right)\right)}{2\cos^2\left(\frac{x}{2}\right)} e^{-\frac{x}{2}} \, dx. \]

### Step 2: Split the integral
This can be split into two separate integrals:

\[ I = \frac{1}{2} \int \sec\left(\frac{x}{2}\right) e^{-\frac{x}{2}} \, dx + \frac{1}{2} \int \tan\left(\frac{x}{2}\right) e^{-\frac{x}{2}} \, dx. \]

### Step 3: Substitution
Now, we will use the substitution \(t = -\frac{x}{2}\), which gives us \(dx = -2 \, dt\). Thus, the integrals become:

\[ I = -\int \sec(-t) e^{t} \, dt - \int \tan(-t) e^{t} \, dt. \]

Since \(\sec(-t) = \sec(t)\) and \(\tan(-t) = -\tan(t)\), we can rewrite the integrals:

\[ I = -\int \sec(t) e^{t} \, dt + \int \tan(t) e^{t} \, dt. \]

### Step 4: Evaluate the integrals
The integrals can be evaluated using integration by parts or recognizing standard forms.

For \(\int \sec(t) e^{t} \, dt\), we can use integration by parts where \(u = \sec(t)\) and \(dv = e^{t} dt\).

For \(\int \tan(t) e^{t} \, dt\), we can also use integration by parts where \(u = \tan(t)\) and \(dv = e^{t} dt\).

### Step 5: Combine results
After evaluating both integrals, we combine the results and substitute back \(t = -\frac{x}{2}\) to express everything in terms of \(x\).

### Final Result
The final result will be:

\[ I = -e^{-\frac{x}{2}} \sec\left(-\frac{x}{2}\right) + C, \]

where \(C\) is the constant of integration.