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Question #40468

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the definite integral: $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left| \sin x - \cos x + \text{sgn}(x) \right| dx$ where $\text{sgn}(x)$ denotes the signum function.

Options

A.
$\pi + 2$
B.
2
C.
$2\sqrt{2}$
D.
$\pi$

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

Geometrical Interpretation of Modulus|Definite Integral Properties (King's Property)|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This problem tests the geometrical interpretation of the modulus function combined with the jump discontinuity of the signum function.

Students often blindly apply symmetry properties (like assuming the function is even, yielding $\pi$ ) or ignore the sign change of the signum function for negative arguments (treating it as a constant 1, which incorrectly evaluates to 2).

Properly splitting the integral at $x=0$ is required because the signum function breaks the continuity, revealing that the integrand behaves fundamentally differently on $[-\pi/2, 0]$ and $[0, \pi/2]$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Geometrical Interpretation of Modulus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free