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

Options

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

Explanation

To solve the integral

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

where \([\cdot]\) denotes the greatest integer function, we will break the integral into two parts: from \(-\frac{\pi}{2}\) to \(0\) and from \(0\) to \(\frac{\pi}{2}\).

### Step 1: Split the Integral
We can write:
\[ I = \int_{-\frac{\pi}{2}}^{0} \frac{dx}{[\text{x}] + [\sin x] + 4} + \int_{0}^{\frac{\pi}{2}} \frac{dx}{[\text{x}] + [\sin x] + 4} \]

### Step 2: Evaluate the First Integral
For the interval \([- \frac{\pi}{2}, 0]\):
- \([\text{x}]\) will be \(-1\) (since \(-1 < x < 0\)).
- \([\sin x]\) will range from \(-1\) to \(0\). Thus, \([\sin x] = -1\) for \(-1 < \sin x < 0\).

So, in this interval:
\[ I_1 = \int_{-\frac{\pi}{2}}^{0} \frac{dx}{-1 - 1 + 4} = \int_{-\frac{\pi}{2}}^{0} \frac{dx}{2} \]
This simplifies to:
\[ I_1 = \frac{1}{2} \left[ x \right]_{-\frac{\pi}{2}}^{0} = \frac{1}{2} \left( 0 - \left(-\frac{\pi}{2}\right) \right) = \frac{\pi}{4} \]

### Step 3: Evaluate the Second Integral
For the interval \([0, \frac{\pi}{2}]\):
- \([\text{x}]\) will be \(0\) (since \(0 \leq x < 1\)).
- \([\sin x]\) will range from \(0\) to \(1\). Thus, \([\sin x] = 0\) for \(0 < \sin x < 1\).

So, in this interval:
\[ I_2 = \int_{0}^{\frac{\pi}{2}} \frac{dx}{0 + 0 + 4} = \int_{0}^{\frac{\pi}{2}} \frac{dx}{4} \]
This simplifies to:
\[ I_2 = \frac{1}{4} \left[ x \right]_{0}^{\frac{\pi}{2}} = \frac{1}{4} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{8} \]

### Step 4: Combine the Results
Now, we can combine both parts:
\[ I = I_1 + I_2 = \frac{\pi}{4} + \frac{\pi}{8} \]

### Step 5: Find a Common Denominator
To add these fractions, we need a common denominator:
\[ \frac{\pi}{4} = \frac{2\pi}{8} \]
Thus,
\[ I = \frac{2\pi}{8} + \frac{\pi}{8} = \frac{3\pi}{8} \]

### Final Answer
The value of the integral is:
\[ \boxed{\frac{3\pi}{8}} \]