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Question #41476

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the integral $I = \int_{0}^{\sqrt{3}} \frac{1}{(1+x^2)^2} \arcsin\left(\frac{2x}{1+x^2}\right) dx$

Options

A.
$\frac{7\pi^2}{144} + \frac{\pi\sqrt{3}}{24} - \frac{1}{8}$
B.
$\frac{\pi^2}{18} + \frac{\pi\sqrt{3}}{12} - \frac{3}{8}$
C.
$\frac{7\pi^2}{72} + \frac{\pi\sqrt{3}}{12} - \frac{1}{4}$
D.
$\frac{5\pi^2}{144} + \frac{\pi\sqrt{3}}{24} - \frac{1}{8}$

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

The Sign-Consistency Principle|Algebraic-Trigonometric Graphical Mapping|Variable Transformation Substitution

Why This Question Is Tricky

Common Mistakes

This question tests the Sign-Consistency Principle and domain awareness during variable transformations.

A common trap is blindly applying the identity $\arcsin(\sin v) = v$ across the entire integration interval.

However, the substitution maps the upper limit into a region ( $v > \pi/2$ ) where the sine function is non-monotonic, requiring a branch change to $\pi - v$ .

Failing to split the integral at the critical point $v = \pi/2$ leads to an incorrect evaluation of the area under the curve (Option B), specifically testing the student's rigor in handling inverse trigonometric principal values.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Sign-Consistency Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free