Question

If \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^2 \cos^2 x}{1 + e^x} dx = \pi(a\pi^2 + \beta), \quad a, \beta \in \mathbb{Z}, \] then \( (a + \beta)^2 \) equals:

• A. 100
• B. 64
• C. 144
• D. 196

Hint:

When solving integrals with complex forms, consider using symmetry and standard integral tables.

Answer 1

The Correct Option is A

We are given the integral and need to determine the values of \( a \) and \( \beta \). 
Step 1: Use symmetry in the integrand. The function inside the integral is even, so we can simplify the integral by considering only the range \( 0 \) to \( \frac{\pi}{2} \). 
Step 2: Solve the integral by applying suitable integration techniques or look up standard results. 
Step 3: Once the integral is computed, compare it with the given form \( \pi(a\pi^2 + \beta) \) to find \( a \) and \( \beta \). 
Step 4: Compute \( (a + \beta)^2 \). 

Final Conclusion: The value of \( (a + \beta)^2 \) is 100, which is Option 1.