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Question #41490

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: (0, e] \to \mathbb{R}$ be a continuous function such that $f(x) + x f'(x) = \begin{cases} 3x^2 & \text{for } 0 < x \le 1 \\ 5 & \text{for } 1 < x \le e \end{cases}$ for all $x \in (0, e] \setminus \{1\}$ . If $\lim_{x \to 0^+} x f(x) = 0$ , then the value of $3 \int_0^e f(x) dx$ is

Options

A.
$15e - 26$
B.
$15e - 14$
C.
$15e - 12$
D.
$15e - 20$

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

Substitution (Change of Variable)|Integrating f(x) + xf'(x) forms|Differentiability vs Continuity

Why This Question Is Tricky

Common Mistakes

The question tests the application of the product rule reverse-substitution d/dx[x f(x)] = f(x) + x f'(x) in the context of a piecewise continuous function.

The primary trap lies in the transition point x=1.

Students often integrate the piecewise branches independently, implicitly assuming the constants of integration are zero (yielding f(x) = 5 for x > 1).

However, because f(x) is continuous, the antiderivative x f(x) must also be continuous, forcing a non-zero constant of integration (C_2 = -4).

This highlights the critical difference between global continuity and piecewise differentiability, punishing the blind application of indefinite integration formulas.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Substitution (Change of Variable)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free