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Question #41873

Difficulty: 8/10 | 🟢 Free

The Question

Let $[t]$ denote the greatest integer less than or equal to $t$ . Evaluate the integral: $I = \int_{1 \le |x| \le 5/3} \left[ \left| x - \sqrt{x^2-1} \right| \right] \frac{(x + \sqrt{x^2-1})^3 - (x - \sqrt{x^2-1})^3}{\sqrt{x^2-1}} \, dx$

Options

A.
$\frac{9493}{648}$
B.
$-\frac{4967}{216}$
C.
$\frac{9493}{324}$
D.
$\frac{1352}{81}$

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

Integral Part and Fractional Part|Newton-Leibniz Formula with variable limits|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of piecewise functions defined by the Greatest Integer Function (GIF) and Modulus, combined with a binomial expansion.

The primary trap is assuming symmetry or ignoring the modulus.

For $x \in [1, 5/3]$ , the GIF evaluates to 0, annihilating the integral on the positive side.

For $x \in [-5/3, -1]$ , the modulus converts the negative argument to positive, shifting the GIF values.

Failing to apply the modulus results in integrating $[-u]$ , which yields negative step values and leads directly to the lethal distractor $-4967/216$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Integral Part and Fractional Part
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free