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Question #41458

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_p(m)$ denote the exponent of the highest power of a prime $p$ dividing $m!$ , and let $S_n = \sum_{k=1}^n k^3$ . Consider the function $f(x)$ defined on $(0, 3)$ by: $f(x) = \sum_{n=1}^{\infty} \frac{x^n}{(2 E_3(3^n!) + 1) \sqrt{S_n}}$ Evaluate the improper integral $\int_0^3 f(x) \, dx$ .

Options

A.
$12 - \pi^2$
B.
$6 - \frac{\pi^2}{2}$
C.
$12 - \frac{\pi^2}{3}$
D.
$8 - \frac{2\pi^2}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Improper Integrals|Sum of Powers of Natural Numbers|Exponent of Prime in n!

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of number theory (Legendre's formula), sequences, and improper integrals.

The primary trap lies in attempting to find a closed-form expression for $f(x)$ before integrating.

While $f(x)$ can be evaluated as $2 + \frac{2(3-x)}{x}\ln(1-\frac{x}{3})$ , integrating this closed form requires a highly error-prone integration by parts involving dilogarithms.

The 'lethal collision' is bypassed by recognizing that swapping the integral and the infinite sum (justified by the Monotone Convergence Theorem for positive terms) transforms a difficult continuous improper integral into a straightforward discrete telescoping and p-series sum.

Additionally, students often miscalculate the geometric progression in Legendre's formula, leading to incorrect coefficients.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Improper Integrals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free