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:
Show Hint
- 100
- 64
- 144
- 196
The Correct Option is A
Solution and Explanation
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.
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