`int[sin^2((9pi)/8+x/4)-sin^2((7pi)/8+x/4)]dx`

Answer

To solve the integral

\[ I = \int \left( \sin^2\left(\frac{9\pi}{8} + \frac{x}{4}\right) - \sin^2\left(\frac{7\pi}{8} + \frac{x}{4}\right) \right) dx, \]

we will use the identity for sine squared, which states:

\[ \sin^2 x = \frac{1 - \cos(2x)}{2}. \]

### Step 1: Apply the sine squared identity
Using the identity, we can rewrite the integral as follows:

\[ I = \int \left( \frac{1 - \cos\left(2\left(\frac{9\pi}{8} + \frac{x}{4}\right)\right)}{2} - \frac{1 - \cos\left(2\left(\frac{7\pi}{8} + \frac{x}{4}\right)\right)}{2} \right) dx. \]

This simplifies to:

\[ I = \int \left( \frac{1}{2} - \frac{\cos\left(\frac{9\pi}{4} + \frac{x}{2}\right)}{2} - \frac{1}{2} + \frac{\cos\left(\frac{7\pi}{4} + \frac{x}{2}\right)}{2} \right) dx. \]

### Step 2: Simplify the integral
Now we can combine the terms:

\[ I = \int \left( \frac{1}{2} \left( \cos\left(\frac{7\pi}{4} + \frac{x}{2}\right) - \cos\left(\frac{9\pi}{4} + \frac{x}{2}\right) \right) \right) dx. \]

### Step 3: Use the cosine subtraction formula
Using the formula for the difference of cosines:

\[ \cos a - \cos b = -2 \sin\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right), \]

let \( a = \frac{7\pi}{4} + \frac{x}{2} \) and \( b = \frac{9\pi}{4} + \frac{x}{2} \).

Calculating \( a + b \) and \( a - b \):

\[ a + b = \frac{7\pi}{4} + \frac{9\pi}{4} + x = 4\pi + x, \]
\[ a - b = \frac{7\pi}{4} - \frac{9\pi}{4} = -\frac{\pi}{2}. \]

Now substituting back into the integral:

\[ I = -\int \sin\left(2\pi + \frac{x}{2}\right) \sin\left(-\frac{\pi}{4}\right) dx. \]

Since \(\sin(-\frac{\pi}{4}) = -\frac{1}{\sqrt{2}}\), we have:

\[ I = \frac{1}{\sqrt{2}} \int \sin\left(2\pi + \frac{x}{2}\right) dx. \]

### Step 4: Integrate
The integral of \(\sin(kx)\) is \(-\frac{1}{k} \cos(kx)\), so:

\[ I = \frac{1}{\sqrt{2}} \left(-\frac{1}{\frac{1}{2}} \cos\left(2\pi + \frac{x}{2}\right)\right) + C = -2\frac{1}{\sqrt{2}} \cos\left(2\pi + \frac{x}{2}\right) + C. \]

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

\[ I = -\frac{2}{\sqrt{2}} \cos\left(2\pi + \frac{x}{2}\right) + C. \]