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Question #41513

Difficulty: 8/10 | 🟢 Free

The Question

Let $F(x) = \int_{-x}^{x} \left( |2t - x| \cos(t) + \frac{t^3 \sin^2(x)}{1 + t^2} \right) dt$ for $x > 0$ . Find the value of $F'(\pi)$ .

Options

A.
$2 - 4\pi$
B.
$-4\pi$
C.
$2\pi$
D.
$2 + 2\pi$

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

Leibniz Rule for Differentiation under Integral|Even and Odd Function Symmetry|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Leibniz rule for differentiation under the integral sign when the integrand contains a modulus function dependent on the boundary variable.

The primary trap is the 'Modulus Sign Dependency': students must correctly evaluate the partial derivative of the modulus term by splitting the integral at the critical point $t = x/2$ .

Failing to account for the signum function or ignoring the modulus leads to incorrect partial integrals (e.g., yielding $-4\pi$ ).

Additionally, an odd function symmetry trap is included as a red herring; students who fail to recognize that the second term vanishes will waste time on complex, fruitless differentiation.

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