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Question #41181

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{x \in [0, 4\pi] : \sin x \ge |\cos x|\}$ . Evaluate the integral $I = \int_{S} x \left| \cos^{-1}(\cos x) - \sin^{-1}(\sin x) \right| dx$

Options

A.
$\frac{5\pi^3}{24}$
B.
$\frac{7\pi^3}{24}$
C.
$\frac{\pi^3}{4}$
D.
$\frac{\pi^3}{6}$

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

Inverse Function Periodicity|Inequalities in Definite Integrals|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' between modulus-restricted domains and the periodicity of inverse trigonometric functions.

Students must first correctly solve the trigonometric inequality to find the disjoint intervals of integration.

The core trap lies in evaluating $\cos^{-1}(\cos x) - \sin^{-1}(\sin x)$ .

Students frequently assume this evaluates to $0$ everywhere or fail to identify the exact transition points (e.g., $\pi/2$ and $5\pi/2$ ) where the principal value branches shift, causing the integrand to become non-zero.

The modulus further penalizes sign errors if the branches are incorrectly identified.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Inverse Function Periodicity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free