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Question #41169

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ . Evaluate the integral $I = \int_{1/\sqrt{3}}^{\sqrt{3}} f(x) \, dx$

Options

A.
$\frac{7\pi\sqrt{3}}{9} + 2\ln\left(\frac{2}{3}\right)$
B.
$\frac{10\pi\sqrt{3}}{9} - 2\ln 3$
C.
$\frac{2\pi\sqrt{3}}{3}$
D.
$\frac{7\pi\sqrt{3}}{9} + 2\ln\left(\frac{3}{2}\right)$

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

Variable Transformation Subsitution|Argument and Principal Value|Principal Value Branch Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding the principal value branch boundaries of inverse trigonometric functions.

A common trap is mechanically applying the identity $\sin^{-1}(\sin \alpha) = \alpha$ without checking if $\alpha \in [-\pi/2, \pi/2]$ .

Here, the substitution $x = \tan \theta$ maps the integration interval such that the argument $2\theta$ crosses the boundary $\pi/2$ at $x=1$ .

Failing to split the integral and apply the correct branch $\sin^{-1}(\sin 2\theta) = \pi - 2\theta$ for $x > 1$ leads to the incorrect trap answer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Variable Transformation Subsitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free