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Question #41163

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral: $I = \int_{-\pi/2}^{3\pi/2} \frac{2|\sin^{-1}(\sin x)| + 4|\cos^{-1}(\cos x) - \frac{\pi}{2}|}{1 + 2024^{\cos^3 x}} \, dx$

Options

A.
$\frac{3\pi^2}{2}$
B.
$3\pi^2$
C.
$\frac{5\pi^2}{4}$
D.
$\frac{3\pi^2}{4}$

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

The Quadrant Adjustment Logic|Definite Integral Properties (King's Property)|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Modulus Trap' within a definite integral requiring King's Property.

Students commonly err by assuming $\sin^{-1}(\sin x) = x$ globally, ignoring principal value branches.

The modulus forces the integrand to be $\pi$ -periodic, requiring careful quadrant adjustment.

By extending limits to $[-\pi/2, 3\pi/2]$ , it tests the ability to recognize periodicity and correctly partition the integral to evaluate the true area, avoiding domain errors.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	The Quadrant Adjustment Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free