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Question #41165

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^{\cos^{-1}(\cos x)} \left( \lfloor t \rfloor + \left\lfloor \frac{2t}{\pi} \right\rfloor + \left\lfloor \frac{3t}{\pi} \right\rfloor \right) dt$ , where $\lfloor \cdot \rfloor$ denotes the greatest integer function. Find the value of $f'\left( \frac{22\pi}{7} \right)$ .

Options

A.
$-5$
B.
$5$
C.
$24$
D.
$8$

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

Composition Inverse Identity Traps|Newton-Leibniz Formula with variable limits|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula on a definite integral with a composite inverse trigonometric function as its upper limit.

The primary trap lies in the assumption that $\cos^{-1}(\cos x) = x$ , which leads students to ignore the piecewise nature of the inverse identity and its derivative.

A secondary trap is forgetting the chain rule multiplier $u'(x) = -1$ in the interval $(3\pi, 4\pi)$ , leading to a sign error.

The inclusion of the Greatest Integer Function (GIF) ensures that evaluating the integrand at the incorrect boundary (e.g., $x$ instead of $4\pi - x$ ) yields a completely different integer, heavily penalizing a global, non-piecewise approach.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Composition Inverse Identity Traps
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free