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Question #41569

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{4}{x} (-1)^{[x]}$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . If $I = \int_{1/3}^{3} f(x) dx$ , then the value of $e^I$ is:

Options

A.
$\frac{6561}{256}$
B.
6561
C.
$\frac{256}{6561}$
D.
$\frac{1}{256}$

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

Region Ambiguity|Rectangular Hyperbola|Step Function Discontinuities|The Sign-Consistency Principle

Why This Question Is Tricky

Common Mistakes

This question tests the 'Sign-Consistency Principle' and 'Region Ambiguity' when integrating a piecewise function defined by the Greatest Integer Function $[x]$ alongside a rectangular hyperbola $xy=c^2$ .

The primary trap lies in the first interval $(1/3, 1)$ , where students frequently misidentify $[x]$ or mishandle the logarithmic evaluation of the lower limit, incorrectly computing $4(0 - \ln(1/3))$ as $-4\ln 3$ instead of $+4\ln 3$ .

This sign error cascades, leading to the trap answer $1/256$ .

Furthermore, completely ignoring the step function's discontinuities yields the distractor $6561$ .

The correct resolution requires strict partitioning at integer bounds and precise sign management.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Region Ambiguity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free