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Question #41422

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(\theta) = \int_0^{\pi/2} \ln(1 - \cos^2\theta \sin^2 x) \, dx$ for $\theta \in (0, 2\pi)$ . Find the value of $f''\left(\frac{7\pi}{6}\right)$ .

Options

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

Check Your Answer

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

Trigonometric Identical Substitutions|Leibniz Rule for Differentiation under Integral|Logarithmic Differentiation Domain|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Leibniz integration and domain/range nuances.

When evaluating the integral $\int_0^\infty \frac{dt}{1 + \sin^2\theta t^2}$ , students frequently write $\frac{\pi}{2\sin\theta}$ instead of the rigorously correct $\frac{\pi}{2|\sin\theta|}$ .

Because $\theta = \frac{7\pi}{6}$ lies in the third quadrant where sine is negative, omitting the absolute value leads to a fundamentally incorrect derivative function $f'(\theta) = \pi(\sec\theta - \tan\theta)$ , resulting in the trap answer $-2\pi$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Trigonometric Identical Substitutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free