`int (px^(p+2q-1)-qx^(q-1))/(x^(2p+2q)+2x^(p+q)+1) dx` is equal to
(1)   `-x^p/(x^(p+q)+1)+C`  
  (2)   `x^q/(x^(p+q)+1)+C`   
(3)   `-x^q/(x^(p+q)+1)+C`   
(4)   `x^p/(x^(p+q)+1)+C`   

Options

A. ` -(x^(p))/(x^(p+q)+1)+C`
B. ` (x^(q))/(x^(p+q)+1)+C`
C. ` -(x^(q))/(x^(p+q)+1)+C`
D. ` (x^(p))/(x^(p+q)+1)+C`

Explanation

To solve the integral

\[ I = \int \frac{px^{p + 2q - 1} - qx^{q - 1}}{x^{2p + 2q} + 2x^{p + q} + 1} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
We can simplify the integral by dividing both the numerator and the denominator by \(x^{2q}\):

\[ I = \int \frac{p x^{p + 2q - 1}/x^{2q} - q x^{q - 1}/x^{2q}}{x^{2p + 2q}/x^{2q} + 2x^{p + q}/x^{2q} + 1/x^{2q}} \, dx. \]

This gives us:

\[ I = \int \frac{p x^{p - 2q - 1} - q x^{-q - 1}}{x^{2p} + 2x^{p - q} + x^{-2q}} \, dx. \]

### Step 2: Rewrite the Denominator
The denominator can be rewritten as:

\[ x^{2p} + 2x^{p - q} + x^{-2q} = (x^p + x^{-q})^2. \]

### Step 3: Substitute
Let \(t = x^p + x^{-q}\). Then, differentiating both sides gives:

\[ dt = (p x^{p - 1} - q x^{-q - 1}) \, dx. \]

### Step 4: Substitute in the Integral
Now, we can rewrite our integral in terms of \(t\):

\[ I = \int \frac{dt}{t^2}. \]

### Step 5: Integrate
The integral of \(t^{-2}\) is:

\[ I = -\frac{1}{t} + C. \]

### Step 6: Substitute Back
Substituting back for \(t\):

\[ I = -\frac{1}{x^p + x^{-q}} + C. \]

### Step 7: Final Form
This can be rewritten as:

\[ I = -\frac{x^q}{x^{p + q} + 1} + C. \]

### Conclusion
Thus, the final answer is:

\[ I = -\frac{x^q}{x^{p + q} + 1} + C. \]