`int(sinx)/(1+sin x) dx`

Answer

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

### Step 1: Multiply and Divide by \( 1 - \sin x \)
We start by multiplying and dividing the integrand by \( 1 - \sin x \):

\[ I = \int \frac{\sin x (1 - \sin x)}{(1 + \sin x)(1 - \sin x)} \, dx \]

### Step 2: Simplify the Denominator
Using the difference of squares, we simplify the denominator:

\[ 1 + \sin x)(1 - \sin x) = 1 - \sin^2 x = \cos^2 x \]

Thus, the integral becomes:

\[ I = \int \frac{\sin x (1 - \sin x)}{\cos^2 x} \, dx \]

### Step 3: Expand the Numerator
Now, we can expand the numerator:

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

### Step 4: Split the Integral
We can split the integral into two parts:

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

### Step 5: Rewrite the Integrals
We can rewrite the integrals using trigonometric identities:

\[ I = \int \tan x \sec^2 x \, dx - \int \tan^2 x \, dx \]

### Step 6: Integrate the First Part
The integral of \( \tan x \sec^2 x \) is:

\[ \int \tan x \sec^2 x \, dx = \sec x \]

### Step 7: Integrate the Second Part
For the second integral, we use the identity \( \tan^2 x = \sec^2 x - 1 \):

\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx \]

The integral of \( \sec^2 x \) is \( \tan x \) and the integral of \( 1 \) is \( x \):

\[ \int \tan^2 x \, dx = \tan x - x \]

### Step 8: Combine the Results
Putting it all together, we have:

\[ I = \sec x - (\tan x - x) + C \]

Thus, simplifying gives:

\[ I = \sec x - \tan x + x + C \]

### Final Answer
The final result for the integral is:

\[ \int \frac{\sin x}{1 + \sin x} \, dx = \sec x - \tan x + x + C \]