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Question #41425

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_{0}^{\sin x} \left( \arcsin t - \frac{\pi}{6} \right) dt$ . Find the value of $\int_{0}^{2\pi} |f'(x)| dx$ .

Options

A.
$2\pi - 2\sqrt{3}$
B.
$\frac{4\pi}{3} - \sqrt{3}$
C.
$2\pi - \sqrt{3}$
D.
$\frac{10\pi}{3} - \sqrt{3} + 2$

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

Trigonometric Identical Substitutions|Algebraic-Trigonometric Graphical Mapping|Critical Point vs Stationary Point|Chain Rule Complexity

Why This Question Is Tricky

Common Mistakes

This question tests the conflation of local stationarity with global extremality within a restricted domain defined by trig-substitution.

Trap 156 (Chain Rule) forces a Jacobian-style error if the student ignores the differential mapping of the domain.

Trap 263 exploits the tendency to solve $f'(x)=0$ by incorrectly assuming $\arcsin(\sin x) = x$ for all $x \in [0, 2\pi]$ , which would cause them to miss the critical point at $x = 5\pi/6$ .

Missing this point fails to catch a sign change in $f'(x)$ , leading to an incorrect definite integral value.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Trigonometric Identical Substitutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free