The value of the integral `I=int_(0)^( pi/4)[sin x+cos x](cos x-sin x)dx` is equal to (where,[.] denotes the greatest integer function)`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{4}} (\sin x + \cos x)(\cos x - \sin x) \, dx \), we will follow these steps:

### Step 1: Expand the integrand
We start by expanding the integrand:
\[ I = \int_{0}^{\frac{\pi}{4}} (\sin x + \cos x)(\cos x - \sin x) \, dx \]
Using the distributive property:
\[ = \int_{0}^{\frac{\pi}{4}} (\sin x \cos x - \sin^2 x + \cos^2 x - \sin x \cos x) \, dx \]
This simplifies to:
\[ = \int_{0}^{\frac{\pi}{4}} (\cos^2 x - \sin^2 x) \, dx \]

### Step 2: Use the identity for cosine and sine
Recall that \( \cos^2 x - \sin^2 x = \cos(2x) \). Thus, we can rewrite the integral:
\[ I = \int_{0}^{\frac{\pi}{4}} \cos(2x) \, dx \]

### Step 3: Integrate
The integral of \( \cos(2x) \) is:
\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C \]
Now, we evaluate the definite integral:
\[ I = \left[ \frac{1}{2} \sin(2x) \right]_{0}^{\frac{\pi}{4}} \]

### Step 4: Evaluate the limits
Calculating the limits:
\[ = \frac{1}{2} \sin\left(2 \cdot \frac{\pi}{4}\right) - \frac{1}{2} \sin(0) \]
\[ = \frac{1}{2} \sin\left(\frac{\pi}{2}\right) - 0 \]
\[ = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Step 5: Apply the greatest integer function
The value of \( I \) is \( \frac{1}{2} \). The greatest integer function \( [I] \) is:
\[ [I] = [\frac{1}{2}] = 0 \]

### Final Answer
Thus, the value of the integral \( I \) is equal to:
\[ \boxed{0} \]