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Question #40407

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral: $I = \int_{-1}^{1} \text{Im}\left( e^{i\pi/3} \frac{1 + i\sqrt{x^2}}{1 - i\sqrt{x^2}} \right) dx$

Options

A.
$\ln 2 + \sqrt{3}\left(\frac{\pi}{2} - 1\right)$
B.
$\sqrt{3}\left(\frac{\pi}{2} - 1\right)$
C.
$2\ln 2 + \sqrt{3}\left(\frac{\pi}{2} - 1\right)$
D.
$\ln 2 + \sqrt{3}\left(\frac{\pi}{2} + 1\right)$

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

Tangent Half-Angle Substitution|Euler's Form and Rotations|Modulus Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the interaction between complex rotations, tangent half-angle substitutions, and piecewise domain constraints.

The complex fraction acts as a tangent half-angle substitution generating a unit complex number.

The trap lies in the term $\sqrt{x^2}$ , which must be evaluated as $|x|$ .

Students who blindly substitute $\sqrt{x^2} = x$ or ignore the modulus during the complex expansion will generate an odd term $\frac{x}{1+x^2}$ .

Over the symmetric interval $[-1, 1]$ , this odd term incorrectly integrates to zero, leading to Option B.

Recognizing the modulus forces the term to be even, contributing a crucial $\ln 2$ to the final integral and invalidating standard continuous algebraic expansions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Tangent Half-Angle Substitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free