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Question #41549

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the definite integral: $I = \int_{0}^{2\pi} \frac{x \sqrt{1 - \cos(2x)}}{\sqrt{2} (1 + \cos^2 x)} dx$

Options

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

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

Symmetry Utilization|Definite Integral Properties (King's Property)|The Quadrant Adjustment Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance with 'Quadrant Adjustment Logic' and domain-restricted simplifications.

A common conceptual reflex is to blindly simplify $\sqrt{1 - \cos(2x)}$ as $\sqrt{2}\sin x$ instead of the mathematically rigorous $\sqrt{2}|\sin x|$ .

If the absolute value is ignored, applying King's Property ( $x \to 2\pi - x$ ) introduces a false sign flip because $\sin(2\pi - x) = -\sin x$ .

This traps the student into the erroneous equation $I = -I$ , resulting in a zero-integral conclusion ( $0$ ).

Recognizing the absolute value preserves the even symmetry of the trigonometric part across the shifted quadrant, allowing King's Property to correctly eliminate the algebraic $x$ multiplier and yield the true non-zero result.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Symmetry Utilization
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free