`intsqrt((1+sinx)/(1-sinx))dx=`

Options

A. `tanx-secx+c`
B. `cosx+cotx+c`
C. `log[sin((pi)/(4)-(x)/(2))]+c`
D. `-log(1-sinx)+c`

Answer

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \sqrt{\frac{1 + \sin x}{1 - \sin x}} \, dx \]

### Step 2: Simplify the Expression
We can simplify the expression under the square root:
\[ \sqrt{\frac{1 + \sin x}{1 - \sin x}} = \frac{\sqrt{1 + \sin x}}{\sqrt{1 - \sin x}} \]

### Step 3: Multiply by a Conjugate
To simplify further, we can multiply the numerator and the denominator by \(\sqrt{1 + \sin x}\):
\[ \frac{\sqrt{1 + \sin x}}{\sqrt{1 - \sin x}} \cdot \frac{\sqrt{1 + \sin x}}{\sqrt{1 + \sin x}} = \frac{1 + \sin x}{\sqrt{(1 - \sin x)(1 + \sin x)}} \]
This simplifies to:
\[ \frac{1 + \sin x}{\sqrt{1 - \sin^2 x}} = \frac{1 + \sin x}{\cos x} \]

### Step 4: Rewrite the Integral
Now we can rewrite our integral:
\[ \int \frac{1 + \sin x}{\cos x} \, dx \]

### Step 5: Split the Integral
We can split the integral into two parts:
\[ \int \frac{1}{\cos x} \, dx + \int \frac{\sin x}{\cos x} \, dx \]
This gives us:
\[ \int \sec x \, dx + \int \tan x \, dx \]

### Step 6: Integrate Each Part
Now we integrate each part:
1. The integral of \(\sec x\) is \(\ln |\sec x + \tan x| + C\).
2. The integral of \(\tan x\) is \(-\ln |\cos x| + C\).

Combining these results, we have:
\[ \int \sec x \, dx + \int \tan x \, dx = \ln |\sec x + \tan x| - \ln |\cos x| + C \]

### Step 7: Simplify the Result
Using the properties of logarithms, we can combine the logarithms:
\[ \ln \left(\frac{\sec x + \tan x}{\cos x}\right) + C \]

### Step 8: Further Simplification
Since \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\), we can rewrite:
\[ \frac{\sec x + \tan x}{\cos x} = \frac{\frac{1}{\cos x} + \frac{\sin x}{\cos x}}{\cos x} = \frac{1 + \sin x}{\cos^2 x} \]

Thus, the final result is:
\[ \ln \left(1 + \sin x\right) + C \]

### Final Answer
The final answer is:
\[ \int \sqrt{\frac{1 + \sin x}{1 - \sin x}} \, dx = \ln \left(1 - \sin x\right) + C \]