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Question #41472

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = e^{[x]} \int_{0}^{x} \text{sgn}(\sin(\pi t)) dt$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Evaluate the definite integral $\int_{0}^{202} f'(x) dx$ , where the integration is performed over the points of differentiability of $f(x)$ .

Options

A.
$0$
B.
$\frac{1 - e^{202}}{1 + e}$
C.
$\frac{e^{202} - 1}{e - 1}$
D.
$\frac{1 + e^{202}}{1 + e}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

The Sign-Consistency Principle|Newton-Leibniz Formula with variable limits|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize the breakdown of the Fundamental Theorem of Calculus (Newton-Leibniz) when a function contains jump discontinuities induced by the Greatest Integer Function.

Students incorrectly assume that $\int_{0}^{202} f'(x) dx = f(202) - f(0) = 0$ .

To solve correctly, they must partition the integral at integer points, apply the Sign-Consistency principle to evaluate the piecewise constant sign of $\sin(\pi t)$ , and sum the resulting geometric progression.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Sign-Consistency Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free