`int_(0)^(pi//4) sin(x-[x])` dx is equalto

Options

A. `(1)/(2)`
B. `1-(1)/(sqrt(2))`
C. 1
D. none of these

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{4}} \sin(x - [x]) \, dx \), we can follow these steps:

### Step 1: Understand the function \( [x] \)
The function \( [x] \) denotes the greatest integer function (or floor function), which gives the largest integer less than or equal to \( x \).

### Step 2: Analyze the limits of integration
Since we are integrating from \( 0 \) to \( \frac{\pi}{4} \) and \( \frac{\pi}{4} \approx 0.785 \), which is less than \( 1 \), we can conclude that for all \( x \) in the interval \( [0, \frac{\pi}{4}] \), the value of \( [x] \) will be \( 0 \).

### Step 3: Simplify the integral
Given that \( [x] = 0 \) in the interval, we can rewrite the integral:
\[ \int_{0}^{\frac{\pi}{4}} \sin(x - [x]) \, dx = \int_{0}^{\frac{\pi}{4}} \sin(x - 0) \, dx = \int_{0}^{\frac{\pi}{4}} \sin(x) \, dx \]

### Step 4: Integrate \( \sin(x) \)
The integral of \( \sin(x) \) is:
\[ -\cos(x) \]
Thus, we evaluate:
\[ \int_{0}^{\frac{\pi}{4}} \sin(x) \, dx = \left[-\cos(x)\right]_{0}^{\frac{\pi}{4}} \]

### Step 5: Apply the limits
Now we substitute the limits into the integrated function:
\[ -\cos\left(\frac{\pi}{4}\right) - (-\cos(0)) = -\left(\frac{1}{\sqrt{2}}\right) + 1 \]
This simplifies to:
\[ 1 - \frac{1}{\sqrt{2}} \]

### Final Result
Thus, the value of the integral \( \int_{0}^{\frac{\pi}{4}} \sin(x - [x]) \, dx \) is:
\[ 1 - \frac{1}{\sqrt{2}} \]

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