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Question #40476

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_n(x) = \int_0^{2\pi} \frac{e^{x \cos\theta} \cos(x \sin\theta + n\theta)}{13 - 12\cos\theta} d\theta$ for $n \in \mathbb{N} \cup \{0\}$ . If $S(x) = \sum_{n=0}^{\infty} I_n(x)$ , find the value of $S\left(\frac{3}{2} \ln 5\right)$ .

Options

A.
$4\pi$
B.
$6\pi$
C.
$8\pi$
D.
$10\pi$

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

Euler's Form and Rotations|Leibniz Rule for Differentiation under Integral|Symmetry Utilization

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of complex rotations and the Leibniz rule for differentiation under the integral sign.

The primary trap lies in 'Symmetry Utilization': students might attempt to expand the trigonometric terms or use standard substitutions, leading to a massive algebraic computation trap.

The elegant path requires mapping the base integral ( $n=0$ ) to a closed-loop contour via Euler's form $z = e^{i\theta}$ to utilize the vanishing integral property of analytic functions outside the poles.

Furthermore, recognizing that the integrand for $I_n(x)$ is the $n$ -th partial derivative of the base integrand with respect to $x$ allows the application of the Leibniz rule in reverse, transforming an infinite series of complex integrals into a simple geometric progression of derivatives.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Euler's Form and Rotations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free