`int(sinx)/((1+sinx))dx=?`

Options

A. `x+tanx-secx+C`
B. `x-tanx-secx+C`
C. `x-tanx+secx+C`
D. none of these

Explanation

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

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

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

### Step 2: Expand the numerator
Now, we can expand the numerator:

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

And the denominator simplifies to:

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

Thus, we can rewrite the integral as:

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

### Step 3: Separate the integrals
Now we can separate the integral into two parts:

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

### Step 4: Rewrite the integrals
The first integral can be rewritten using the identity \( \frac{\sin x}{\cos^2 x} = \tan x \sec x \):

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

### Step 5: Integrate the first term
The integral of \( \tan x \sec x \) is:

\[ \int \tan x \sec x \, dx = \sec x + C \]

### Step 6: Integrate the second term
For the second integral, we can 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 \( x \):

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

### Step 7: Combine the results
Now we can combine the results of our integrals:

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

Thus, the final result is:

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

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