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Question #40473

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_{-1}^{1} x \cdot \text{Arg}(x^2 - 1 + 2ix) \, dx$ , where $\text{Arg}(z)$ denotes the principal argument of $z$ such that $\text{Arg}(z) \in (-\pi, \pi]$ . Find the value of $I$ .

Options

A.
2
B.
$2 - \pi$
C.
$2 + \pi$
D.
0

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

Argument and Principal Value|Integration by Parts (LIATE Rule)|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle the principal value of complex arguments during integration.

A common trap is blindly applying logarithmic or arctangent properties, such as assuming $\text{Arg}(z^2) = 2\text{Arg}(z)$ universally or using $\tan^{-1}(\frac{2x}{x^2-1})$ without quadrant analysis.

Here, $z = x+i$ has an argument of $\cot^{-1}(x)$ .

For $x < 0$ , $2\text{Arg}(x+i)$ exceeds $\pi$ , requiring a $-2\pi$ phase shift to remain in the principal branch $(-\pi, \pi]$ .

Ignoring this branch cut restriction leads to the incorrect primitive and the trap answer $2-\pi$ .

The problem demands rigorous domain analysis before applying Integration by Parts.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Argument and Principal Value
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free