`int_(pi)^((3pi)/2)(sin^(4)x+cos^(4)x)dx`

Answer

To solve the integral

\[ I = \int_{\pi}^{\frac{3\pi}{2}} \left( \sin^4 x + \cos^4 x \right) dx, \]

we can follow these steps:

### Step 1: Rewrite the integrand
We can use the identity for \(a^4 + b^4\):

\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x. \]

Since \(\sin^2 x + \cos^2 x = 1\), we have:

\[ \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x. \]

### Step 2: Substitute into the integral
Now we substitute this back into the integral:

\[ I = \int_{\pi}^{\frac{3\pi}{2}} \left( 1 - 2\sin^2 x \cos^2 x \right) dx. \]

### Step 3: Simplify the integral
We can separate the integral:

\[ I = \int_{\pi}^{\frac{3\pi}{2}} 1 \, dx - 2 \int_{\pi}^{\frac{3\pi}{2}} \sin^2 x \cos^2 x \, dx. \]

### Step 4: Calculate the first integral
The first integral is straightforward:

\[ \int_{\pi}^{\frac{3\pi}{2}} 1 \, dx = \left[ x \right]_{\pi}^{\frac{3\pi}{2}} = \frac{3\pi}{2} - \pi = \frac{\pi}{2}. \]

### Step 5: Calculate the second integral
For the second integral, we use the identity:

\[ \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x). \]

Thus, we have:

\[ \int_{\pi}^{\frac{3\pi}{2}} \sin^2 x \cos^2 x \, dx = \frac{1}{4} \int_{\pi}^{\frac{3\pi}{2}} \sin^2(2x) \, dx. \]

### Step 6: Use the identity for \(\sin^2\)
We can use the identity:

\[ \sin^2(2x) = \frac{1 - \cos(4x)}{2}. \]

So,

\[ \int_{\pi}^{\frac{3\pi}{2}} \sin^2(2x) \, dx = \frac{1}{2} \int_{\pi}^{\frac{3\pi}{2}} (1 - \cos(4x)) \, dx. \]

### Step 7: Calculate the integral
Now we can compute:

\[ \int_{\pi}^{\frac{3\pi}{2}} 1 \, dx = \frac{\pi}{2}, \]

and

\[ \int_{\pi}^{\frac{3\pi}{2}} \cos(4x) \, dx = \left[ \frac{\sin(4x)}{4} \right]_{\pi}^{\frac{3\pi}{2}} = \frac{\sin(6\pi)}{4} - \frac{\sin(4\pi)}{4} = 0. \]

Thus,

\[ \int_{\pi}^{\frac{3\pi}{2}} \sin^2(2x) \, dx = \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4}. \]

### Step 8: Substitute back
Now substituting back into our expression for \(I\):

\[ I = \frac{\pi}{2} - 2 \cdot \frac{1}{4} \cdot \frac{\pi}{4} = \frac{\pi}{2} - \frac{\pi}{8} = \frac{4\pi}{8} - \frac{\pi}{8} = \frac{3\pi}{8}. \]

### Final Answer
Thus, the value of the integral is:

\[ \boxed{\frac{3\pi}{8}}. \]