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Question #41427

Difficulty: 8/10 | 🟢 Free

The Question

Let $D$ be the maximal subset of $\mathbb{R}$ for which the function $f(x) = \sqrt{5 + 4x - x^2} - \sqrt{x^2 - 6x + 9}$ is real-valued. The value of the integral $\int_{D} x f(x) \,dx$ is

Options

A.
$9\pi - \frac{34}{3}$
B.
$9\pi - \frac{17}{3}$
C.
$\frac{9\pi}{2} - \frac{34}{3}$
D.
$9\pi + \frac{34}{3}$

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

Completing the Square in Quadratics|Parabola Shifted Vertex|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the 'Hidden Bound' trap where the limits of integration are not explicitly given but must be derived from the domain of a shifted quadratic inside a square root.

It further challenges students with the modulus logic $\sqrt{u^2} = |u|$ , requiring them to split the integral at the shifted vertex.

Failing to recognize the modulus, ignoring the domain restriction, or incorrectly shifting the vertex during the substitution $x-2=t$ are common pitfalls designed to filter out rote learners.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Completing the Square in Quadratics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free