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Question #40412

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_{0}^{\pi} \frac{\sin\left(2x + \frac{\pi}{6}\right) \cos\left(x + \frac{\pi}{6}\right) - \cos\left(2x + \frac{\pi}{6}\right) \sin\left(x + \frac{\pi}{6}\right)}{3 + \cos(2x)} \left\{ \frac{2x}{\pi} \right\} \, dx$ where $\{t\}$ denotes the fractional part of $t$ . Find the value of $I$ .

Options

A.
$\frac{\pi}{8}$
B.
$\frac{\pi}{4}$
C.
$\frac{\pi}{2}$
D.
0

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Compound Angle Expansion|Definite Integral Properties (King's Property)|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of compound angle identities with definite integral properties, specifically targeting a common trap involving the fractional part function.

Students often incorrectly assume that $\{-y\} = -\{y\}$ or ignore the fractional part's piecewise nature, which would lead to an answer of 0 or $\pi/4$ .

By applying King's Property, the argument of the fractional part transforms to $2 - \frac{2x}{\pi}$ .

Recognizing that $\{n - y\} = 1 - \{y\}$ for non-integers is crucial to eliminate the fractional part by adding the original and transformed integrals, bypassing a tedious piecewise evaluation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Compound Angle Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free