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Question #41405

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n(x) = \frac{1369}{2} \left( n - \frac{\sin(nx)\cos((n+1)x)}{\sin x} \right)$ for $x \neq m\pi$ , where $m \in \mathbb{Z}$ . A sequence of functions is defined as $T_n(x) = S_n(x) - S_{n-1}(x)$ for $n \ge 2$ , and $U_n(x) = \sqrt{T_n(x)}$ . Evaluate the integral $I = \int_0^{\pi/4} U_{10}(x) dx$

Options

A.
$\frac{37}{2}$
B.
$\frac{37}{10}$
C.
$37$
D.
$\frac{185}{4}$

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

Linearity and Partitioning|Sum to n terms vs. General Term|Square Root and Modulus Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to extract a general term from a summation formula and handle the critical $\sqrt{x^2} = |x|$ trap in calculus.

By defining $S_n(x)$ as a closed-form trigonometric sum, students must first deduce $T_n(x) = 1369 \sin^2(nx)$ .

The primary filter occurs when taking the square root: failing to include the absolute value results in integrating $37 \sin(10x)$ directly, leading to the trap answer $37/10$ .

The correct path requires recognizing the modulus, identifying the periodicity of $|\sin(10x)|$ , and partitioning the integral over $2.5$ periods to obtain the true area.

Quick Facts About This Question

Question Information

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