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Question #41413

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral: $I = \int_{-3\pi}^{3\pi} \frac{|x \cos x|}{1 + 2024^x} \, dx$

Options

A.
$9\pi$
B.
$6\pi$
C.
$\frac{19\pi}{2}$
D.
$10\pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Integration by Parts (LIATE Rule)|Even and Odd Function Symmetry|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate a modulus-wrapped product of algebraic and trigonometric functions.

The primary trap lies in attempting to apply Integration by Parts (LIATE) directly to $|x \cos x|$ , which is invalid because the modulus creates points of non-differentiability at the roots of $\cos x$ .

Students must first recognize the even symmetry to eliminate the exponential denominator, then meticulously decompose the integral into sub-intervals where the function is smooth, before finally applying IBP.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integration by Parts (LIATE Rule)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free