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

Concept: The energy of a hydrogen electron is \(E_n = -13.6/n^2\) eV. The excitation energy is the difference between the final and initial states. Step 1: Calculate level energies. Ground state (\(n=1\)): \(E_1 = -13.6\) eV. First excited state (\(n=2\)): \(E_2 = -13.6/4 = -3.4\) eV.
Step 2: Find the difference. \(\Delta E = E_2 - E_1 = -3.4 - (-13.6) = 10.2\) eV.
Step 3: Convert to Joules. Using \(1\,\text{eV} \approx 1.6 \times 10^{-19}\,\text{J}\): \[ \Delta E = 10.2 \times (1.602 \times 10^{-19}) \approx 1.634 \times 10^{-18}\,\text{J} \] \[ \boxed{\Delta E = 1.634\times 10^{-18}\ \text{J}} \]