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Question #41531

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that for all $x \neq 0$ , $\int_0^1 f(xt) dt = f(x) - x f'(x) + \left(x^2 + 1 - \frac{1}{x}\right) e^x + \frac{1}{x}$ If $f(1) = e + 2$ , then the value of $f(2)$ is:

Options

A.
$2e^2 + 2$
B.
$2e^2$
C.
$2e^2 + \ln 2 + 2$
D.
$2e^2 + 2e$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Integrating f(x) + xf'(x) forms|Leibniz Rule for Differentiation under Integral|Arbitrary Constants vs Parameters

Why This Question Is Tricky

Common Mistakes

The question tests the application of the Leibniz rule to an integral equation where the parameter is hidden in the upper limit after substitution.

The core anchor is recognizing $x f''(x) + f'(x)$ as the exact derivative of $x f'(x)$ .

The primary trap lies in the 'Arbitrary Constants vs Parameters' concept: students often ignore the domain constraint ( $f: \mathbb{R} \to \mathbb{R}$ ) and carry an arbitrary constant $C$ , leading to incorrect logarithmic terms, or they incorrectly assume all integration constants are zero (yielding $2e^2$ ).

The condition $f(1) = e+2$ specifically isolates the second constant $K$ , rewarding only those who correctly distinguish between the boundary conditions for $C$ (implicit via differentiability at $x=0$ ) and $K$ (explicit).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integrating f(x) + xf'(x) forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free