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Question #41436

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \begin{vmatrix} \int_0^x \sin t \, dt & \int_0^x \cos t \, dt & \int_0^x e^{t^2} \, dt \\ \sin x & \cos x & e^{x^2} \\ \cos x & -\sin x & 2x e^{x^2} \end{vmatrix}$ . Find the value of $\lim_{x \to 0} \frac{f(x)}{x^3}$ .

Options

A.
$-\frac{1}{2}$
B.
$\frac{1}{2}$
C.
$-\frac{3}{2}$
D.
0

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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

Leibniz Rule for Differentiation under Integral|Determinant as Geometric Area/Volume|Linear Dependency of Rows|Properties of Transpose Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Leibniz rule for differentiation under the integral sign extended to determinants.

The primary trap lies in the presence of the non-elementary integral $\int_0^x e^{t^2} dt$ , which intentionally derails students who attempt to expand the determinant algebraically before differentiating.

The elegant bypass requires recognizing that differentiating the determinant row-by-row generates matrices where the first two have linearly dependent (identical) rows, causing their determinants to vanish identically.

This reduces a seemingly impossible expansion into a single, manageable determinant, rewarding structural algebraic insight over brute-force calculation.

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