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Question #41480

Difficulty: 8/10 | 🟢 Free

The Question

Consider the improper iterated integrals over the unbounded region $[1, \infty) \times [1, \infty)$ : $I_1 = \int_1^\infty \left( \int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2} \, dy \right) \, dx$ $I_2 = \int_1^\infty \left( \int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2} \, dx \right) \, dy$ Which of the following represents the ordered pair $(I_1, I_2)$ ?

Options

A.
$\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$
B.
$\left(\frac{\pi}{4}, -\frac{\pi}{4}\right)$
C.
(0, 0)
D.
$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

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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

Axis of Integration Selection|Bounded vs Unbounded Regions|Improper Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of Fubini's Theorem and improper integrals over unbounded regions.

A common trap is assuming that the order of integration can be swapped freely (yielding $I_1 = I_2$ ) or that anti-symmetry implies both integrals are zero.

However, Fubini's Theorem requires the double integral of the absolute value of the function to be finite.

Here, $\iint |f(x,y)| \, dx \, dy$ diverges over the given unbounded region.

Consequently, the iterated integrals yield different, non-zero results depending entirely on the chosen axis of integration, forcing a rigorous limit-based evaluation for each path.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Axis of Integration Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free