Before You Begin

Question #41453

Difficulty: 8/10 | 🟢 Free

The Question

Consider the definite integrals: $I_1 = \left| \int_0^{2\pi} \left( e^{i \arcsin(\sin x)} - e^{i \arccos(\cos x)} \right) dx \right|$ $I_2 = \int_0^{2\pi} \left| e^{i \arcsin(\sin x)} - e^{i \arccos(\cos x)} \right| dx$ where $i = \sqrt{-1}$ . Find the ordered pair $(I_1, I_2)$ .

Options

A.
$(0, 0)$
B.
$(4\sqrt{2}, 4 + \pi)$
C.
$(4, 4)$
D.
$(4\sqrt{2}, 4\sqrt{2})$

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

Inequalities in Definite Integrals|Euler's Form and Rotations|The Quadrant Adjustment Logic

Why This Question Is Tricky

Common Mistakes

This question tests the 'Quadrant Adjustment Logic' trap within the context of complex definite integrals and integral inequalities.

Students often incorrectly assume $e^{i \arcsin(\sin x)} = e^{ix}$ by ignoring the principal branch restrictions of inverse trigonometric functions.

By forcing the evaluation of both the modulus of the integral and the integral of the modulus, the problem demands rigorous application of $\cos(\arcsin y) = \sqrt{1-y^2}$ and $\sin(\arccos y) = \sqrt{1-y^2}$ , which naturally introduces absolute values ( $|\cos x|$ and $|\sin x|$ ).

Failing to account for these phase shifts across quadrants leads to the trap answer $(0,0)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Inequalities in Definite Integrals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free