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Question #41466

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be the area of the region bounded by the curve $y = \frac{\sin x}{\cos^3 x} \ln\left| \frac{\cos 2x}{\cos^2 x} \right|$ the x-axis, and the lines $x = 0$ and $x = \frac{\pi}{3}$ . The value of $A$ is:

Options

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

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 Simplification|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle definite integrals of logarithmic functions where the argument undergoes sign changes and magnitude fluctuations, deploying the 'Wavy Curve' trap.

Students must first algebraically simplify the trigonometric argument into a polynomial form ( $1 - \tan^2 x$ ).

The critical trap lies in evaluating the area, which requires integrating the absolute value of the function.

Students commonly ignore the absolute value around the logarithm or fail to recognize that $\ln|v|$ changes sign at $v = \pm 1$ .

If they blindly integrate without splitting the domain at the roots of the logarithm, they will incorrectly compute the net signed area ( $\ln 2 - 1.5$ ) instead of the true geometric area, leading directly to the trap option.

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