Solve the following integration
`int (cos x - sin x)/(cos x + sin x)*(2+2 sin 2x)dx`

Answer

To solve the integral

\[ \int \frac{\cos x - \sin x}{\cos x + \sin x} (2 + 2 \sin 2x) \, dx, \]

we will follow these steps:

### Step 1: Simplify the integrand
First, we can factor out the constant from the second term in the integrand:

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

### Step 2: Rewrite \(\sin 2x\)
Recall that \(\sin 2x = 2 \sin x \cos x\). Thus, we can rewrite the integrand as:

\[ 2 \int \frac{\cos x - \sin x}{\cos x + \sin x} \left(1 + 2 \sin x \cos x\right) \, dx. \]

### Step 3: Expand the integrand
Now, we can distribute the terms in the integrand:

\[ = 2 \int \left(\frac{\cos x - \sin x}{\cos x + \sin x} + \frac{2(\cos x - \sin x) \sin x \cos x}{\cos x + \sin x}\right) \, dx. \]

### Step 4: Simplify the first term
The first term simplifies to:

\[ \frac{\cos x - \sin x}{\cos x + \sin x}. \]

### Step 5: Simplify the second term
For the second term, we can observe that:

\[ \frac{2(\cos x - \sin x) \sin x \cos x}{\cos x + \sin x} = \frac{2 \sin x \cos x (\cos x - \sin x)}{\cos x + \sin x}. \]

### Step 6: Use the identity
Notice that \(1 + \sin 2x = 1 + 2 \sin x \cos x\) can be rewritten using the identity \(1 + \sin 2x = \cos^2 x + \sin^2 x + 2 \sin x \cos x = (\cos x + \sin x)^2\).

### Step 7: Substitute in the integral
Now we can substitute this back into our integral:

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

This simplifies to:

\[ = 2 \int (\cos x - \sin x)(\cos x + \sin x) \, dx. \]

### Step 8: Apply the difference of squares
Using the difference of squares, we have:

\[ = 2 \int (\cos^2 x - \sin^2 x) \, dx. \]

### Step 9: Use the double angle identity
Using the identity \(\cos^2 x - \sin^2 x = \cos 2x\):

\[ = 2 \int \cos 2x \, dx. \]

### Step 10: Integrate
The integral of \(\cos 2x\) is:

\[ = 2 \cdot \frac{1}{2} \sin 2x + C = \sin 2x + C. \]

### Final Answer
Thus, the final answer is:

\[ \int \frac{\cos x - \sin x}{\cos x + \sin x} (2 + 2 \sin 2x) \, dx = \sin 2x + C. \]