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Question #41470

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sum_{n=1}^{\infty} \tan^{-1}\left( \frac{1}{1 + (x+n)(x+n-1)} \right)$ . Evaluate the integral: $I = \int_{-1}^{-\frac{1}{\sqrt{3}}} \frac{f(x) + x f'(x)}{|x| f(x)} \, dx$

Options

A.
$\ln\left( \frac{9\sqrt{3}}{8} \right)$
B.
$\ln\left( \frac{8}{9\sqrt{3}} \right)$
C.
$\ln\left( \frac{3\sqrt{3}}{4} \right)$
D.
$\ln\left( \frac{4}{3\sqrt{3}} \right)$

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

Integrating f(x) + xf'(x) forms|Logarithmic Integration Modulus|Infinite Series involving Inverse Tangent

Why This Question Is Tricky

Common Mistakes

The question tests two major JEE Advanced concepts: evaluating an infinite telescoping inverse trigonometric series and handling absolute values in logarithmic integration.

The primary trap lies in the definition of $|x|$ for negative domains; students who blindly integrate the numerator as the exact derivative of the denominator to get $\ln|u|$ without accounting for $|x| = -x$ will obtain the negative of the correct answer (Option B).

A secondary trap involves incorrectly evaluating the series sum as $\tan^{-1}(1/x)$ instead of $\cot^{-1}(x)$ , which leads to incorrect boundary values for negative $x$ (Options C and D).

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