Question

The value of \( \int_{-\pi/2}^{\pi/2} \frac{dx}{[x]+4} \) is, where [.] denotes greatest integer function:

• A. \(\frac{7\pi}{20} - \frac{7}{60}\)
• B. \(\frac{\pi}{20} - \frac{11}{30}\)
• C. \(\frac{11\pi}{20} - \frac{7}{30}\)
• D. \(\frac{11\pi}{30} - \frac{7}{20}\)

Hint:

When integrating a function involving the greatest integer function \([f(x)]\), always split the integral at the points where the argument \(f(x)\) takes integer values.

Answer 1

The Correct Option is A

Step 1: Understanding the Question and Splitting the Integral:
We need to evaluate a definite integral containing the greatest integer function, \([x]\). The function \([x]\) is a step function which changes its value at every integer. Therefore, we must split the integral at all integer points within the integration interval \([-\pi/2, \pi/2]\).
The approximate value of \(\pi/2\) is 1.57. The integers in the interval \([-1.57, 1.57]\) are -1, 0, and 1.
So we split the integral as follows: \[ I = \int_{-\pi/2}^{\pi/2} \frac{dx}{[x]+4} = \int_{-\pi/2}^{-1} \frac{dx}{[x]+4} + \int_{-1}^{0} \frac{dx}{[x]+4} + \int_{0}^{1} \frac{dx}{[x]+4} + \int_{1}^{\pi/2} \frac{dx}{[x]+4} \]
Step 2: Evaluating the Greatest Integer Function in Each Interval:

For \(x \in [-\pi/2, -1)\) (i.e., \(-1.57 \leq x<-1\)), we have \([x] = -2\).
For \(x \in [-1, 0)\), we have \([x] = -1\).
For \(x \in [0, 1)\), we have \([x] = 0\).
For \(x \in [1, \pi/2]\) (i.e., \(1 \leq x \leq 1.57\)), we have \([x] = 1\).
Step 3: Calculating the Integral:
Now substitute these constant values into the integrals: \[ I = \int_{-\pi/2}^{-1} \frac{dx}{-2+4} + \int_{-1}^{0} \frac{dx}{-1+4} + \int_{0}^{1} \frac{dx}{0+4} + \int_{1}^{\pi/2} \frac{dx}{1+4} \] \[ I = \int_{-\pi/2}^{-1} \frac{1}{2} dx + \int_{-1}^{0} \frac{1}{3} dx + \int_{0}^{1} \frac{1}{4} dx + \int_{1}^{\pi/2} \frac{1}{5} dx \] Now, we perform the integration: \[ I = \frac{1}{2}[x]_{-\pi/2}^{-1} + \frac{1}{3}[x]_{-1}^{0} + \frac{1}{4}[x]_{0}^{1} + \frac{1}{5}[x]_{1}^{\pi/2} \] \[ I = \frac{1}{2}(-1 - (-\pi/2)) + \frac{1}{3}(0 - (-1)) + \frac{1}{4}(1 - 0) + \frac{1}{5}(\pi/2 - 1) \] \[ I = \frac{1}{2}(-1 + \pi/2) + \frac{1}{3}(1) + \frac{1}{4}(1) + \frac{1}{5}(\pi/2 - 1) \]
Step 4: Final Calculation:
Group the terms with \(\pi\) and the constant terms: \[ I = (-\frac{1}{2} + \frac{\pi}{4}) + \frac{1}{3} + \frac{1}{4} + (\frac{\pi}{10} - \frac{1}{5}) \] \[ I = \left(\frac{\pi}{4} + \frac{\pi}{10}\right) + \left(-\frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5}\right) \] The \(\pi\) part: \[ \frac{5\pi + 2\pi}{20} = \frac{7\pi}{20} \] The constant part: \[ \frac{-30 + 20 + 15 - 12}{60} = \frac{-7}{60} \] So, the calculated value of the integral is \(I = \frac{7\pi}{20} - \frac{7}{60}\).