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Question #41501

Difficulty: 8/10 | 🟢 Free

The Question

Let $I = \int_{-\sqrt{3}}^{\sqrt{3}} \frac{|x| \sqrt{1+x^2}}{x^4+1} dx$ . By substituting $x = \tan \theta$ and subsequently $t = \sec \theta$ , the integral $I$ is transformed into the form $2 \int_{1}^{2} g(t) dt$ . Let $M$ be the maximum value of $g(t)$ for $t \in [1, 2]$ . Find the number of integers $k \in [-5, 5]$ satisfying the inequality: $\frac{k^2 - 2k}{k - 3} \le (2M - \sqrt{2})(k^2 - 2k)$

Options

A.
7
B.
9
C.
10
D.
8

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Trigonometric Identical Substitutions|Arithmetic Mean Inequality|Cross Multiplication Trap|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Function Breakdown' by forcing students to split the integral at $x=0$ before applying the trigonometric substitution $x = \tan \theta$ , ensuring the bounds and signs are handled correctly.

It then evaluates the 'AM-GM Inequality' to find the global maximum of the rationalized integrand's denominator.

Finally, it deploys the classic 'Cross Multiplication Trap' in the resulting algebraic inequality.

Students who carelessly divide by the variable expression $(k^2 - 2k)$ or cross-multiply without considering the sign of $(k-3)$ will lose critical intervals (such as $k \le 0$ or $k=2$ ) or incorrectly include invalid points (like $k=1$ ), leading to an incorrect integer count.

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