`int(1)/(1-sinx) dx`

Answer

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

### Step 1: Rationalize the Denominator
We start with the integral:
\[ \int \frac{1}{1 - \sin x} \, dx \]
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + \sin x\):
\[ \int \frac{1}{1 - \sin x} \cdot \frac{1 + \sin x}{1 + \sin x} \, dx \]
This simplifies to:
\[ \int \frac{1 + \sin x}{(1 - \sin x)(1 + \sin x)} \, dx \]

### Step 2: Simplify the Denominator
Using the identity \(a^2 - b^2 = (a - b)(a + b)\), we can simplify the denominator:
\[ (1 - \sin x)(1 + \sin x) = 1 - \sin^2 x = \cos^2 x \]
Thus, the integral becomes:
\[ \int \frac{1 + \sin x}{\cos^2 x} \, dx \]

### Step 3: Split the Integral
Now we can separate the integral into two parts:
\[ \int \frac{1}{\cos^2 x} \, dx + \int \frac{\sin x}{\cos^2 x} \, dx \]
This can be written as:
\[ \int \sec^2 x \, dx + \int \tan x \sec x \, dx \]

### Step 4: Integrate Each Part
1. The integral of \(\sec^2 x\) is:
\[ \int \sec^2 x \, dx = \tan x \]
2. The integral of \(\tan x \sec x\) is:
\[ \int \tan x \sec x \, dx = \sec x \]

### Step 5: Combine the Results
Combining both results, we have:
\[ \tan x + \sec x + C \]
where \(C\) is the constant of integration.

### Final Answer
Thus, the final answer for the integral \(\int \frac{1}{1 - \sin x} \, dx\) is:
\[ \tan x + \sec x + C \]

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