The value of `int_(-pi//2)^(pi//2)dx/([x]+[sinx]+4)` where [t] denotes the greatest integer less or equal to t, is

Options

A. `1/12(7pi+5)`
B. `3/10(4pi-3)`
C. `1/12(7pi-5)`
D. `3/20(4pi-3)`

Answer

To solve the integral

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{dx}{[x] + [\sin x] + 4} \]

where \([t]\) denotes the greatest integer less than or equal to \(t\), we will break the integral into segments where the greatest integer function remains constant.

### Step 1: Identify the points where the greatest integer function changes
The limits of integration are from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). We need to identify the integer points in this range where the nature of the function changes. The integer points in this interval are \(-1\), \(0\), and \(1\).

### Step 2: Break the integral into segments
We can break the integral into four parts based on the identified points:

\[ I = \int_{-\frac{\pi}{2}}^{-1} \frac{dx}{[x] + [\sin x] + 4} + \int_{-1}^{0} \frac{dx}{[x] + [\sin x] + 4} + \int_{0}^{1} \frac{dx}{[x] + [\sin x] + 4} + \int_{1}^{\frac{\pi}{2}} \frac{dx}{[x] + [\sin x] + 4} \]

### Step 3: Evaluate each segment
1. **For \(x \in \left[-\frac{\pi}{2}, -1\right)\)**:
- \([x] = -2\)
- \([\sin x] = -1\) (since \(\sin x\) ranges from \(-1\) to \(0\))
- Thus, the integral becomes:
\[ \int_{-\frac{\pi}{2}}^{-1} \frac{dx}{-2 - 1 + 4} = \int_{-\frac{\pi}{2}}^{-1} \frac{dx}{1} = \left[-x\right]_{-\frac{\pi}{2}}^{-1} = -(-1) + \frac{\pi}{2} = 1 + \frac{\pi}{2} \]

2. **For \(x \in [-1, 0)\)**:
- \([x] = -1\)
- \([\sin x] = -1\)
- Thus, the integral becomes:
\[ \int_{-1}^{0} \frac{dx}{-1 - 1 + 4} = \int_{-1}^{0} \frac{dx}{2} = \frac{1}{2} \left[x\right]_{-1}^{0} = \frac{1}{2}(0 - (-1)) = \frac{1}{2} \]

3. **For \(x \in [0, 1)\)**:
- \([x] = 0\)
- \([\sin x] = 0\)
- Thus, the integral becomes:
\[ \int_{0}^{1} \frac{dx}{0 + 0 + 4} = \int_{0}^{1} \frac{dx}{4} = \frac{1}{4} \left[x\right]_{0}^{1} = \frac{1}{4}(1 - 0) = \frac{1}{4} \]

4. **For \(x \in [1, \frac{\pi}{2}]\)**:
- \([x] = 1\)
- \([\sin x] = 0\)
- Thus, the integral becomes:
\[ \int_{1}^{\frac{\pi}{2}} \frac{dx}{1 + 0 + 4} = \int_{1}^{\frac{\pi}{2}} \frac{dx}{5} = \frac{1}{5} \left[x\right]_{1}^{\frac{\pi}{2}} = \frac{1}{5}\left(\frac{\pi}{2} - 1\right) \]

### Step 4: Combine all parts
Now we combine all the evaluated integrals:

\[ I = \left(1 + \frac{\pi}{2}\right) + \frac{1}{2} + \frac{1}{4} + \frac{1}{5}\left(\frac{\pi}{2} - 1\right) \]

### Step 5: Simplify the expression
Combining the constants and the terms involving \(\pi\):

1. The constant terms:
\[ 1 + \frac{1}{2} + \frac{1}{4} - \frac{1}{5} = \frac{20}{20} + \frac{10}{20} + \frac{5}{20} - \frac{4}{20} = \frac{31}{20} \]

2. The \(\pi\) terms:
\[ \frac{\pi}{2} + \frac{\pi}{10} = \frac{5\pi}{10} + \frac{\pi}{10} = \frac{6\pi}{10} = \frac{3\pi}{5} \]

Thus, the final answer is:

\[ I = \frac{31}{20} + \frac{3\pi}{5} \]

### Final Answer
\[ I = \frac{31 + 12\pi}{20} \]