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Question #41462

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral: $I = \int_{0}^{\pi} \left( \left[ \log_2 \left( \frac{32}{5 - 3\cos x} \right) \right] + \left[ \log_2 \left( \frac{32}{5 + 3\cos x} \right) \right] \right) dx$ where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
$4\pi + 2\arccos\left(\frac{1}{3}\right)$
B.
$5\pi$
C.
$2\pi + \arccos\left(\frac{1}{3}\right)$
D.
$6\pi$

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

Logarithmic Integration via Trig Manipulation|Algebraic-Trigonometric Graphical Mapping|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle the Greatest Integer Function (GIF) in definite integrals alongside logarithmic and trigonometric mappings.

A lethal trap lies in the false assumption that $\int ([A] + [B]) dx = \int [A+B] dx$ .

If a student combines the logarithmic terms before applying the GIF, they evaluate $\int_{0}^{\pi} [\log_2(\frac{1024}{25 - 9\cos^2 x})] dx$ , which incorrectly yields $5\pi$ since the combined argument remains between $40.96$ and $64$ , making its base-2 logarithm strictly between $5.35$ and $6$ .

The correct approach requires recognizing the symmetry via King's Rule ( $x \to \pi - x$ ) to decouple the integral into $2 \int [f(x)] dx$ , and then identifying the exact point of discontinuity where the logarithmic argument hits a power of 2 (at $\cos x = 1/3$ ).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Logarithmic Integration via Trig Manipulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free