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Question #41435

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_{0}^{\pi} \frac{\sin^2 t - \sin^2 x}{|\sin t - \sin x|} dt$ for $x \in \left(0, \frac{\pi}{2}\right)$ . Find the value of $f'\left(\frac{\pi}{4}\right)$ .

Options

A.
$-4\sqrt{2}$
B.
$0$
C.
$-2\sqrt{2}$
D.
$4\sqrt{2}$

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

Leibniz Rule for Differentiation under Integral|Linearity and Partitioning|Domain Constraint Intersections|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Leibniz Rule for differentiation under the integral sign when the integrand contains a moving singularity.

The 'Denominator Restriction' creates points where the partial derivative is undefined, violating the continuity requirement of the standard Leibniz rule.

Students often fall into the 'Domain Constraint Intersection' trap by blindly moving the derivative inside the integral, which incorrectly yields 0.

The mathematically sound approach requires 'Partitioning' the integral at the singularities $t=x$ and $t=\pi-x$ , revealing hidden boundary terms that contribute $-8\sin x$ to the final derivative.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Leibniz Rule for Differentiation under Integral
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free