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Question #41428

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_{1}^{15} \left( \sum_{k=1}^{[x]} \frac{k^2 \sin^k \left( \frac{\pi \{x\}}{2} \right)}{ \sin^k \left( \frac{\pi \{x\}}{2} \right) + \cos^k \left( \frac{\pi \{x\}}{2} \right) } \right) dx$ where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\{x\}$ denotes the fractional part of $x$ . The value of $I$ is:

Options

A.
2100
B.
4200
C.
2720
D.
1050

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)|Sigma Notation Properties|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

Students often attempt to apply King's Property globally over the entire interval $[1, 15]$ by substituting $x \to 16 - x$ .

However, the presence of the greatest integer function $[x]$ in the summation's upper limit and the fractional part $\{x\}$ introduces periodic jump discontinuities, breaking global symmetry.

The trap forces the student to recognize that piecewise continuity demands partitioning the integral into unit intervals.

This transforms the problem from a pure calculus question into a multi-layered discrete summation series, rigorously testing the interplay between definite integral properties and sequence summation.

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