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Question #41561

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = |x|$ and $g(x) = x|x-2| - 1$ . Evaluate the integral $\int_{0}^{3} f(g(x)) \, dx$

Options

A.
$\frac{8\sqrt{2} - 7}{3}$
B.
1
C.
$\frac{8\sqrt{2} + 1}{3}$
D.
$\frac{4\sqrt{2} - 1}{3}$

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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 Modulus Trap|Composition Continuity|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' within composite functions.

Students often correctly identify the critical point of the inner modulus ( $x=2$ ) but fail to apply the Method of Intervals to the resulting piecewise quadratic $x^2-2x-1$ on $[2, 3]$ .

By neglecting the new root $x=1+\sqrt{2}$ introduced by the composition $f(g(x))$ , they integrate across a sign change, leading to the incorrect answer of 1.

It enforces rigorous case-based decomposition.

Quick Facts About This Question

Question Information

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