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Question #40361

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{e^x \cos x}{x}$ for $x \neq 0$ . Evaluate the integral $I = \int_{-\pi/2}^{\pi/2} \left( \text{sgn}(x) f(x) + x \text{sgn}(x) f'(x) \right) \, dx$ where $\text{sgn}(x)$ denotes the signum function.

Options

A.
-2
B.
0
C.
2
D.
$-e^{\pi/2}$

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

Signum Function Discontinuity|Integrating f(x) + xf'(x) forms|Indeterminate Forms

Why This Question Is Tricky

Common Mistakes

This question tests the student's rigor regarding the Fundamental Theorem of Calculus in the presence of jump discontinuities.

The integrand is carefully constructed as the derivative of $H(x) = x \text{sgn}(x) f(x)$ .

A student who blindly applies the product rule and evaluates $[H(x)]_{-\pi/2}^{\pi/2}$ will incorrectly obtain 0 (Option B).

The 'lethal trap' lies in failing to recognize that $H(x) = \text{sgn}(x) e^x \cos x$ has a jump discontinuity at $x=0$ .

Rigorous evaluation requires splitting the integral at the origin, revealing a non-zero contribution from the boundary limits at the discontinuity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Signum Function Discontinuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free