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Question #41509

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $\int_{-\infty}^{\infty} f(x) dx = 2$ and for all $x \neq 0$ , $f'(x) + \text{sgn}(x) f(x) = e^{-|x|}$ . Let $I = \int_{-\pi/2}^{\pi/2} \frac{(f'(x) - f'(-x)) \sin x + (f(x) + f(-x)) \cos x}{1 + e^x} dx$ If $H(t) = \int_{-1}^{1} |x-t| f'(x) dx$ , find the value of $I \times (H''(0^+) - H''(0^-))$ .

Options

A.
$-8e^{-\pi/2}$
B.
$0$
C.
$-\frac{\pi+2}{2}e^{-\pi/2}$
D.
$-4e^{-\pi/2}$

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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

Definite Integral Properties (King's Property)|Exact Differential Recognition|Integrating Factor (IF) Sign Sensitivity|Non-Differentiable Extrema

Why This Question Is Tricky

Common Mistakes

This question synthesizes exact differentials, King's property, and piecewise differentiability.

The 'Integrating Factor Sign Sensitivity' trap occurs if students ignore $\text{sgn}(x)$ in the differential equation, leading to an incorrect constant of integration $C=1/2$ and a flawed $f(x)$ .

The 'Non-Differentiable Extrema' trap is embedded in $H(t)$ ; students who blindly differentiate without checking the singularity at $x=0$ will incorrectly assume $H''(0^+) = H''(0^-)$ , yielding 0.

Recognizing the exact differential $d((f(x)+f(-x))\sin x)$ and applying King's property on the resulting even function is required to evaluate $I$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Definite Integral Properties (King's Property)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free