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Question #41580

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sqrt{x} |\ln x|$ for $x > 0$ . Let $g(x)$ be the anti-derivative of $f(x)$ such that $g(x)$ is continuous for all $x > 0$ and $g(e^2) = \frac{8e^3}{9}$ . Then the value of $9g(e^{-2})$ is:

Options

A.
$\frac{16}{e^3} - 8$
B.
$\frac{16}{e^3}$
C.
$-\frac{16}{e^3}$
D.
$\frac{16}{e^3} + 8$

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

Compound Function Boundaries|Integration by Parts (LIATE Rule)|Integration Constant Trap|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the 'Integration Constant Trap' in piecewise continuous anti-derivatives.

Students often compute the indefinite integral of a modulus function by simply attaching a single constant of integration $C$ , ignoring the boundary where the modulus changes sign.

Because the modulus forces a split at $x=1$ , the anti-derivative requires two distinct constants, $C_1$ and $C_2$ , to maintain continuity.

Failing to equate the left-hand and right-hand limits at $x=1$ leads directly to the incorrect assumption that $C_1 = C_2 = 0$ , yielding the trap answer $\frac{16}{e^3}$ .

Quick Facts About This Question

Question Information

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