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Question #41508

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_k = \int_0^{\pi/2} \sum_{r=0}^{2k} \binom{k}{r} \left( \frac{\sin^{2r} x}{\sin^{2r} x + \cos^{2r} x} + \frac{r}{k} \right) dx$ for $k \in \mathbb{N}$ . Evaluate the limit: $L = \lim_{n \to \infty} \frac{1}{n^2} \ln \left( \prod_{k=1}^n \frac{2}{\pi} I_k \right)$

Options

A.
$\ln 2$
B.
$\frac{\ln 2}{2}$
C.
$2 \ln 2$
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Definite Integral Properties (King's Property)|Binomial Coefficient Summation Properties|The Empty Set Trap|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question tests the integration of binomial series using King's Property while embedding two lethal traps.

The 'Empty Set Trap' is triggered by the summation limit $2k$ ; students who mechanistically apply summation formulas without realizing $\binom{k}{r} = 0$ for $r > k$ will incorrectly evaluate the series (e.g., assuming the sum yields $2^{2k}$ ), leading to an incorrect limit of $\ln 2$ .

The 'L'Hôpital's Rule Misuse' trap targets students who see the $\infty/\infty$ limit and attempt to differentiate the product of integrals with respect to $n$ .

Since $n$ is discrete and embedded in the binomial upper limits, continuous differentiation is mathematically invalid and practically impossible, forcing the student to first evaluate the series symmetry and the definite integral to a constant expression before taking the limit.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Definite Integral Properties (King's Property)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free